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A177971
Triangle of polynomial coefficients: p(x,n) = (1 - 2x)^(n + 1)*(1 + Sum_{k>=1} 2^(k - 1)*k^n*x^k).
0
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.
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
KEYWORD
sign,tabf,less
AUTHOR
Roger L. Bagula, May 16 2010
STATUS
approved