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, 2, 2, 5, 22, 5, 14, 178, 178, 14, 41, 1308, 3446, 1308, 41, 122, 9234, 52084, 52084, 9234, 122, 365, 64082, 692707, 1434812, 692707, 64082, 365, 1094, 442082, 8559030, 32285474, 32285474, 8559030, 442082, 1094, 3281, 3048184, 101121500, 641507528

Offset

0,2

Comments

Row sums are:

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

3805072588800,...}.

This results from a modular form bilinear approach summed:

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

Links

Table of n, a(n) for n=0..39.

Formula

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

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

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(n,m)=coefficients(p(x,n))

Example

{1},
{2, 2},
{5, 22, 5},
{14, 178, 178, 14},
{41, 1308, 3446, 1308, 41},
{122, 9234, 52084, 52084, 9234, 122},
{365, 64082, 692707, 1434812, 692707, 64082, 365},
{1094, 442082, 8559030, 32285474, 32285474, 8559030, 442082, 1094},
{3281, 3048184, 101121500, 641507528, 1151050534, 641507528, 101121500, 3048184, 3281},
{9842, 21054946, 1161593320, 11747808904, 34632940348, 34632940348, 11747808904, 1161593320, 21054946, 9842},
{29525, 145795662, 13106403569, 203453044136, 928796844218, 1514068354580, 928796844218, 203453044136, 13106403569, 145795662, 29525}

Mathematica

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;
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: A158059 A173323 A019099 * A103890 A292699 A014566

Adjacent sequences: A154644 A154645 A154646 * A154648 A154649 A154650

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, Jan 13 2009

Status

approved