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A154630 A triangular sequence of polynomial coefficients: {a,b,c,d}={4, 5, 5, 0}; p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}]. 0
1, -5, 5, 25, 30, 25, -125, -1145, -775, 125, 625, 17180, 50150, 9500, 625, -3125, -201495, -1596850, -1916750, -155625, 3125, 15625, 2110330, 35871175, 120411500, 70354375, 2256250, 15625, -78125, -20789845, -666806625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums are:

{1, 0, 80, -1920, 78080, -3870720, 231034880, -16078110720, 1278679777280,

-114405691883520, 11373487802286080}

LINKS

Table of n, a(n) for n=0..30.

FORMULA

{a,b,c,d}={4, 5, 5, 0};

p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];

t(n,m)=Coefficients(p(x,n));

p(x,n)=(-4)^n*(1-5*n)^(n+1)*LerchPhi[5*x,-n,5/4]

EXAMPLE

{1},

{-5, 5},

{25, 30, 25},

{-125, -1145, -775, 125},

{625, 17180, 50150, 9500, 625},

{-3125, -201495, -1596850, -1916750, -155625, 3125},

{15625, 2110330, 35871175, 120411500, 70354375,2256250, 15625},

{-78125, -20789845, -666806625, -4737333625, -8100074375, -2518809375, -34296875, 78125},

{390625, 197655480, 11059318300, 143881301000, 525401583750, 508474325000, 89152937500, 511875000, 390625},

{-1953125, -1839446195, -170454627700, -3717803721500, -24702156853750, -52116976906250, -30551523062500, -3137244687500, -7692578125, 1953125},

{9765625, 16896812630, 2501024320325, 86214038005000, 946292721901250, 3649629323762500, 4794767436781250, 1783862843125000, 110088189453125, 115308593750, 9765625}

MATHEMATICA

Clear[p, a, b, c, d, n];

{a, b, c, d} = {4, 5, 5, 0} p[x_, n_] = (-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A074872 A162962 A170834 * A245520 A245130 A263872

Adjacent sequences:  A154627 A154628 A154629 * A154631 A154632 A154633

KEYWORD

sign

AUTHOR

Roger L. Bagula, Jan 13 2009

STATUS

approved

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)