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%I #3 Mar 30 2012 17:34:29
%S 1,-5,5,25,30,25,-125,-1145,-775,125,625,17180,50150,9500,625,-3125,
%T -201495,-1596850,-1916750,-155625,3125,15625,2110330,35871175,
%U 120411500,70354375,2256250,15625,-78125,-20789845,-666806625
%N 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}].
%C Row sums are:
%C {1, 0, 80, -1920, 78080, -3870720, 231034880, -16078110720, 1278679777280,
%C -114405691883520, 11373487802286080}
%F {a,b,c,d}={4, 5, 5, 0};
%F p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];
%F t(n,m)=Coefficients(p(x,n));
%F p(x,n)=(-4)^n*(1-5*n)^(n+1)*LerchPhi[5*x,-n,5/4]
%e {1},
%e {-5, 5},
%e {25, 30, 25},
%e {-125, -1145, -775, 125},
%e {625, 17180, 50150, 9500, 625},
%e {-3125, -201495, -1596850, -1916750, -155625, 3125},
%e {15625, 2110330, 35871175, 120411500, 70354375,2256250, 15625},
%e {-78125, -20789845, -666806625, -4737333625, -8100074375, -2518809375, -34296875, 78125},
%e {390625, 197655480, 11059318300, 143881301000, 525401583750, 508474325000, 89152937500, 511875000, 390625},
%e {-1953125, -1839446195, -170454627700, -3717803721500, -24702156853750, -52116976906250, -30551523062500, -3137244687500, -7692578125, 1953125},
%e {9765625, 16896812630, 2501024320325, 86214038005000, 946292721901250, 3649629323762500, 4794767436781250, 1783862843125000, 110088189453125, 115308593750, 9765625}
%t Clear[p, a, b, c, d, n];
%t {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 Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
%t Flatten[%]
%K sign
%O 0,2
%A _Roger L. Bagula_, Jan 13 2009
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