A177971 Triangle of polynomial coefficients: p(x,n) = (1 - 2x)^(n + 1)*(1 + Sum_{k>=1} 2^(k - 1)*k^n*x^k).

1, -1, 1, -3, 4, 1, -5, 14, -8, 1, -7, 32, -28, 16, 1, -9, 62, -36, 88, -32, 1, -11, 112, 104, 448, -176, 64, 1, -13, 198, 928, 2976, 240, 480, -128, 1, -15, 352, 4316, 20448, 17264, 5632, -960, 256, 1, -17, 638, 16500, 126968, 245872, 142752, 11200, 2432, -512, 1

Offset

0,4

Comments

Row sums are {0, 2, 2, 14, 74, 542, 4682, 47294, 545834, 7087262, 102247562, ...}.

References

G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Translations of Mathematica Monographs, Volume 59, American Mathematical Society, Rhode Island, 1984, pages 9ff.

Links

Table of n, a(n) for n=0..54.

Formula

p(x,n) = (1 - 2x)^(n + 1)*(1 + Sum_{k>=1} 2^(k - 1)*k^n*x^k); t(n,m) = coefficients(t(n,m)).

Example

{1, -1},
{1, -3, 4},
{1, -5, 14, -8},
{1, -7, 32, -28, 16},
{1, -9, 62, -36, 88, -32},
{1, -11, 112, 104, 448, -176, 64},
{1, -13, 198, 928, 2976, 240, 480, -128},
{1, -15, 352, 4316, 20448, 17264, 5632, -960, 256},
{1, -17, 638, 16500, 126968, 245872, 142752, 11200, 2432, -512},
{1, -19, 1184, 57472, 709232, 2490976, 2836928, 919552, 75776, -4864, 1024},
{1, -21, 2246, 190040, 3646816, 20950880, 41960896, 29090048, 6165760, 231168, 11776, -2048}

Mathematica

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

Crossrefs

Sequence in context: A133779 A137911 A019599 * A114156 A348972 A354718

Adjacent sequences: A177968 A177969 A177970 * A177972 A177973 A177974

Keyword

sign,tabf,less

Author

Roger L. Bagula, May 16 2010

Status

approved