A176198 A symmetrical triangle of polynomial coefficients:q=2;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, 11, 1, 1, 45, 45, 1, 1, 151, 459, 151, 1, 1, 473, 3363, 3363, 473, 1, 1, 1443, 21085, 47095, 21085, 1443, 1, 1, 4357, 121313, 519445, 519445, 121313, 4357, 1, 1, 13103, 663223, 4970575, 9350027, 4970575, 663223, 13103, 1, 1, 39345, 3512679

Offset

0,5

Comments

Row sums are:

{1, 2, 13, 92, 763, 7674, 92153, 1290232, 20643831, 371589110, 7431782389,...}.

Links

Table of n, a(n) for n=0..47.

Formula

q=2;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,2)=coefficients(p(x,n,2))

Example

{1},
{1, 1},
{1, 11, 1},
{1, 45, 45, 1},
{1, 151, 459, 151, 1},
{1, 473, 3363, 3363, 473, 1},
{1, 1443, 21085, 47095, 21085, 1443, 1},
{1, 4357, 121313, 519445, 519445, 121313, 4357, 1},
{1, 13103, 663223, 4970575, 9350027, 4970575, 663223, 13103, 1},
{1, 39345, 3512679, 43415943, 138826587, 138826587, 43415943, 3512679, 39345, 1},
{1, 118075, 18232281, 356601807, 1813846563, 3054184935, 1813846563, 356601807, 18232281, 118075, 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: A202678 A202971 A202675 * A202870 A202872 A144440

Adjacent sequences: A176195 A176196 A176197 * A176199 A176200 A176201

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Apr 11 2010

Status

approved