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A154647 A triangular sequence of coefficients: p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(4*m + 3)^n*x^m, {m, 0, Infinity}] + (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(4*m + 1)^n*x^m, {m, 0, Infinity}])/2. 1

%I

%S 1,2,2,5,22,5,14,178,178,14,41,1308,3446,1308,41,122,9234,52084,52084,

%T 9234,122,365,64082,692707,1434812,692707,64082,365,1094,442082,

%U 8559030,32285474,32285474,8559030,442082,1094,3281,3048184,101121500,641507528

%N A triangular sequence of coefficients: p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(4*m + 3)^n*x^m, {m, 0, Infinity}] + (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(4*m + 1)^n*x^m, {m, 0, Infinity}])/2.

%C Row sums are:

%C {1, 4, 32, 384, 6144, 122880, 2949120, 82575360, 2642411520, 95126814720,

%C 3805072588800,...}.

%C This results from a modular form bilinear approach summed:

%C f1(x)=(4*x+1)/(-x); f2(x)=(4*x+3)/(-x).

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

%F (-1)^(-1 + n)* 4^n* (-1 + x)(1 - n) LerchPhi[x, -n, 3/4])/2;

%F p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(4*m + 3)^n*x^m, {m, 0, Infinity}] +

%F (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(4*m + 1)^n*x^m, {m, 0, Infinity}])/2;

%F t(n,m)=coefficients(p(x,n))

%e {1},

%e {2, 2},

%e {5, 22, 5},

%e {14, 178, 178, 14},

%e {41, 1308, 3446, 1308, 41},

%e {122, 9234, 52084, 52084, 9234, 122},

%e {365, 64082, 692707, 1434812, 692707, 64082, 365},

%e {1094, 442082, 8559030, 32285474, 32285474, 8559030, 442082, 1094},

%e {3281, 3048184, 101121500, 641507528, 1151050534, 641507528, 101121500, 3048184, 3281},

%e {9842, 21054946, 1161593320, 11747808904, 34632940348, 34632940348, 11747808904, 1161593320, 21054946, 9842},

%e {29525, 145795662, 13106403569, 203453044136, 928796844218, 1514068354580, 928796844218, 203453044136, 13106403569, 145795662, 29525}

%t Clear[p]; p[x_, n_] = ((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(4*m + 3)^ n*x^m, {m, 0, Infinity}] + (-1)^(n + 1)*(x - 1)^(n + 1)* Sum[(4*m + 1)^n*x^m, {m, 0, Infinity}])/2;

%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 nonn,uned,tabl

%O 0,2

%A _Roger L. Bagula_, Jan 13 2009

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Last modified October 7 16:02 EDT 2022. Contains 357275 sequences. (Running on oeis4.)