A176199 A symmetrical triangle of polynomial coefficients:q=4;p(x,n,q)=(1 - x)^(n + 1)*Sum[((q*k + 1)^n + (q*k + q - 1)^n)*x^k, {k, 0, Infinity}]

1, 1, 1, 1, 35, 1, 1, 329, 329, 1, 1, 2535, 6811, 2535, 1, 1, 18225, 103925, 103925, 18225, 1, 1, 127435, 1384685, 2868895, 1384685, 127435, 1, 1, 881977, 17115873, 64568761, 64568761, 17115873, 881977, 1, 1, 6089807, 202236439, 1283008495

Offset

0,5

Comments

Row sums are:

{1, 2, 37, 660, 11883, 244302, 5893137, 165133224, 5284763991, 190253432610,

7610144528061,...}.

Links

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

Formula

q=4;p(x,n,q)=(1 - x)^(n + 1)*Sum[((q*k + 1)^n + (q*k + q - 1)^n)*x^k, {k, 0, Infinity}];

t(n,m,4)=coefficients(p(x,n,4));

Alternative polynomial function:

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

Example

{1},
{1, 1},
{1, 35, 1},
{1, 329, 329, 1},
{1, 2535, 6811, 2535, 1},
{1, 18225, 103925, 103925, 18225, 1},
{1, 127435, 1384685, 2868895, 1384685, 127435, 1},
{1, 881977, 17115873, 64568761, 64568761, 17115873, 881977, 1},
{1, 6089807, 202236439, 1283008495, 2302094507, 1283008495, 202236439, 6089807, 1},
{1, 42090209, 2323166957, 23495598125, 69265861013, 69265861013, 23495598125, 2323166957, 42090209, 1},
{1, 291532275, 26212748089, 406906029223, 1857593629387, 3028136650111, 1857593629387, 406906029223, 26212748089, 291532275, 1}

Mathematica

p[x_, n_, q_] = (1 - x)^(n + 1)* Sum[((q*k + 1)^n + (q*k + q - 1)^n)*x^k, {k, 0, Infinity}];
f[n_, m_, q_] := CoefficientList[FullSimplify[ExpandAll[p[x, n, q]]], x][[m + 1]];
Table[Flatten[Table[Table[FullSimplify[ ExpandAll[f[ n, m, q] - f[n, 0, q] + 1]], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]

Crossrefs

Cf. A008518, A174599

Sequence in context: A225313 A028847 A365895 * A059023 A327004 A061045

Adjacent sequences: A176196 A176197 A176198 * A176200 A176201 A176202

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Apr 11 2010

Status

approved