A169625 Coefficients of infinite sum polynomials; p(x,n)=If[Mod[n, 2] == 1, (1 - x)^(n + 1)*Sum[(k + 1)*(1 + k + k^2)^Floor[(n - 1)/2]* x^k, {k, 0, Infinity}], (1 - x)^(n + 1)*Sum[(1 + k + k^2)^Floor[n/2]*x^ k, {k, 0, Infinity}]]

1, 1, 1, 0, 1, 1, 2, 3, 1, 4, 14, 4, 1, 1, 12, 54, 44, 9, 1, 20, 175, 328, 175, 20, 1, 1, 46, 625, 2012, 1859, 470, 27, 1, 72, 1708, 9784, 17190, 9784, 1708, 72, 1, 1, 152, 5628, 49384, 134870, 127464, 41308, 3992, 81, 1, 232, 14189, 199616, 884498, 1431728, 884498

Offset

0,7

Comments

Row sums are factorial:

{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800,...}.

Links

Table of n, a(n) for n=0..56.

Formula

p(x,n)=If[Mod[n, 2] == 1, (1 - x)^(n + 1)*Sum[(k + 1)*(1 + k + k^2)^Floor[(n - 1)/2]* x^k, {k, 0, Infinity}],

(1 - x)^(n + 1)*Sum[(1 + k + k^2)^Floor[n/2]*x^ k, {k, 0, Infinity}]]

Example

{1},
{1},
{1, 0, 1},
{1, 2, 3},
{1, 4, 14, 4, 1},
{1, 12, 54, 44, 9},
{1, 20, 175, 328, 175, 20, 1},
{1, 46, 625, 2012, 1859, 470, 27},
{1, 72, 1708, 9784, 17190, 9784, 1708, 72, 1},
{1, 152, 5628, 49384, 134870, 127464, 41308, 3992, 81},
{1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1}

Mathematica

p[x_, n_] = If[Mod[n, 2] == 1, (1 - x)^(n + 1)*Sum[(k + 1)*( 1 + k + k^2)^Floor[(n - 1)/2]*x^k, {k, 0, Infinity}],
(1 - x)^(n + 1)*Sum[(1 + k + k^2)^Floor[n/2]*x^k, {k, 0, Infinity}]]
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]
Flatten[%]

Crossrefs

Sequence in context: A125182 A318685 A270312 * A264794 A264704 A264659

Adjacent sequences: A169622 A169623 A169624 * A169626 A169627 A169628

Keyword

nonn,uned

Author

Roger L. Bagula and Gary W. Adamson, Dec 03 2009

Status

approved