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Question 23066: Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0045-23066 · q-20261014-advance-20261111-r02-higher-order-derivative-0041

Original question and givens

Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

  • G001 — Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source operation=higher-polynomial-nth-derivative;original snapshot and mathematics retained

After

explicit ordered source output fields=[coefficients, order, derivative_coefficients, point, value, certificate]

Conditions: none

Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112];declared order=7

After

declared degree BOUND=6=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Conditions: none

Length minus1 is the declared bound;do not trim trailing zeros.Order0 is the original function.The monomial rule and finite sums justify each polynomial derivative,not an infinite extrapolation for arbitrary functions.

Step identifiers and dependencies

Rule: higher_ORIGINAL_low_to_high_coefficients_degree_BOUND_order0 · Uses: G001, S001

S003 · calculation

Before

Higher derivative level1 ORIGINAL power1 coefficient product := mul(32, 1)

After

Higher derivative level1 ORIGINAL power1 coefficient product = 32

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Higher derivative level1 ORIGINAL power2 coefficient product := mul(48, 2)

After

Higher derivative level1 ORIGINAL power2 coefficient product = 96

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Higher derivative level1 ORIGINAL power3 coefficient product := mul(64, 3)

After

Higher derivative level1 ORIGINAL power3 coefficient product = 192

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Higher derivative level1 ORIGINAL power4 coefficient product := mul(80, 4)

After

Higher derivative level1 ORIGINAL power4 coefficient product = 320

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Higher derivative level1 ORIGINAL power5 coefficient product := mul(96, 5)

After

Higher derivative level1 ORIGINAL power5 coefficient product = 480

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Higher derivative level1 ORIGINAL power6 coefficient product := mul(112, 6)

After

Higher derivative level1 ORIGINAL power6 coefficient product = 672

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

previous derivative order=0;actual low-to-high list=[16, 32, 48, 64, 80, 96, 112];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 32, 1, 32], [2, 48, 2, 96], [3, 64, 3, 192], [4, 80, 4, 320], [5, 96, 5, 480], [6, 112, 6, 672]];next raw list=[32, 96, 192, 320, 480, 672];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S008

S010 · calculation

Before

Higher derivative level2 ORIGINAL power1 coefficient product := mul(96, 1)

After

Higher derivative level2 ORIGINAL power1 coefficient product = 96

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Higher derivative level2 ORIGINAL power2 coefficient product := mul(192, 2)

After

Higher derivative level2 ORIGINAL power2 coefficient product = 384

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Higher derivative level2 ORIGINAL power3 coefficient product := mul(320, 3)

After

Higher derivative level2 ORIGINAL power3 coefficient product = 960

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Higher derivative level2 ORIGINAL power4 coefficient product := mul(480, 4)

After

Higher derivative level2 ORIGINAL power4 coefficient product = 1920

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Higher derivative level2 ORIGINAL power5 coefficient product := mul(672, 5)

After

Higher derivative level2 ORIGINAL power5 coefficient product = 3360

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

previous derivative order=1;actual low-to-high list=[32, 96, 192, 320, 480, 672];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 96, 1, 96], [2, 192, 2, 384], [3, 320, 3, 960], [4, 480, 4, 1920], [5, 672, 5, 3360]];next raw list=[96, 384, 960, 1920, 3360];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S014

S016 · calculation

Before

Higher derivative level3 ORIGINAL power1 coefficient product := mul(384, 1)

After

Higher derivative level3 ORIGINAL power1 coefficient product = 384

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Higher derivative level3 ORIGINAL power2 coefficient product := mul(960, 2)

After

Higher derivative level3 ORIGINAL power2 coefficient product = 1920

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Higher derivative level3 ORIGINAL power3 coefficient product := mul(1920, 3)

After

Higher derivative level3 ORIGINAL power3 coefficient product = 5760

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Higher derivative level3 ORIGINAL power4 coefficient product := mul(3360, 4)

After

Higher derivative level3 ORIGINAL power4 coefficient product = 13440

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

previous derivative order=2;actual low-to-high list=[96, 384, 960, 1920, 3360];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 384, 1, 384], [2, 960, 2, 1920], [3, 1920, 3, 5760], [4, 3360, 4, 13440]];next raw list=[384, 1920, 5760, 13440];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S019

S021 · calculation

Before

Higher derivative level4 ORIGINAL power1 coefficient product := mul(1920, 1)

After

Higher derivative level4 ORIGINAL power1 coefficient product = 1920

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Higher derivative level4 ORIGINAL power2 coefficient product := mul(5760, 2)

After

Higher derivative level4 ORIGINAL power2 coefficient product = 11520

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Higher derivative level4 ORIGINAL power3 coefficient product := mul(13440, 3)

After

Higher derivative level4 ORIGINAL power3 coefficient product = 40320

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

previous derivative order=3;actual low-to-high list=[384, 1920, 5760, 13440];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 1920, 1, 1920], [2, 5760, 2, 11520], [3, 13440, 3, 40320]];next raw list=[1920, 11520, 40320];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S023

S025 · calculation

Before

Higher derivative level5 ORIGINAL power1 coefficient product := mul(11520, 1)

After

Higher derivative level5 ORIGINAL power1 coefficient product = 11520

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Higher derivative level5 ORIGINAL power2 coefficient product := mul(40320, 2)

After

Higher derivative level5 ORIGINAL power2 coefficient product = 80640

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

previous derivative order=4;actual low-to-high list=[1920, 11520, 40320];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 11520, 1, 11520], [2, 40320, 2, 80640]];next raw list=[11520, 80640];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S026

S028 · calculation

Before

Higher derivative level6 ORIGINAL power1 coefficient product := mul(80640, 1)

After

Higher derivative level6 ORIGINAL power1 coefficient product = 80640

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

previous derivative order=5;actual low-to-high list=[11520, 80640];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 80640, 1, 80640]];next raw list=[80640];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S028

S030 · implies

Before

previous derivative order=6;actual low-to-high list=[80640];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S029

S031 · implies

Before

ORIGINAL coefficients=[16, 32, 48, 64, 80, 96, 112];order=7;declared degree bound=6

After

final derivative coefficients=[0];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Conditions: none

Repeated differentiation gives the falling factorial in original coefficient order;orders above the declared bound vanish.This gives no analyticity claim for arbitrary differentiable functions and does not turn a trailing-zero bound into the minimal degree.

Step identifiers and dependencies

Rule: higher_falling_factorial_rule_all_orders_polynomial_only · Uses: G001, S030

S032 · calculation

Before

Higher derivative Horner power0 previous value times ORIGINAL point := mul(0, -2)

After

Higher derivative Horner power0 previous value times ORIGINAL point = 0

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 0)

After

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 0

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

Reverse checks

  • S033 · premise_and_result_audit — Recompute Higher derivative Horner power0 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Higher derivative Horner power0 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Recompute Higher derivative level6 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Recompute Higher derivative level5 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Higher derivative level5 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Recompute Higher derivative level4 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Higher derivative level4 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Higher derivative level4 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power5 coefficient product with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power6 coefficient product with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power5 coefficient product with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 4de15f92aa3c0b467ff9bfc96224a3134c99cc2b47071e84434a67bdea72b106
  • candidate record SHA-256: d177509b25dd7d3cfb4cb33361a95eba2fb3971e89a52bd7cc56ad354f8c2a4b
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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