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[%]
CROSSREFS
KEYWORD
sign
AUTHOR
Roger L. Bagula, Jan 13 2009
STATUS
approved