[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Revision History for A154630 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing all changes.
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}].
(history; published version)
#3 by Russ Cox at Fri Mar 30 17:34:29 EDT 2012
AUTHOR

_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Jan 13 2009

Discussion
Fri Mar 30
17:34
OEIS Server: https://oeis.org/edit/global/158
#2 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010
KEYWORD

nonn,new

sign

#1 by N. J. A. Sloane at Fri Feb 27 03:00:00 EST 2009
NAME

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}].

DATA

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

OFFSET

0,2

COMMENTS

Row sums are:

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

-114405691883520, 11373487802286080}

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[%]

KEYWORD

nonn

AUTHOR

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 13 2009

STATUS

approved