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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

%I #9 Jul 22 2019 14:14:54

%S 1,-1,1,-3,4,1,-5,14,-8,1,-7,32,-28,16,1,-9,62,-36,88,-32,1,-11,112,

%T 104,448,-176,64,1,-13,198,928,2976,240,480,-128,1,-15,352,4316,20448,

%U 17264,5632,-960,256,1,-17,638,16500,126968,245872,142752,11200,2432,-512,1

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

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

%D 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.

%F 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)).

%e {1, -1},

%e {1, -3, 4},

%e {1, -5, 14, -8},

%e {1, -7, 32, -28, 16},

%e {1, -9, 62, -36, 88, -32},

%e {1, -11, 112, 104, 448, -176, 64},

%e {1, -13, 198, 928, 2976, 240, 480, -128},

%e {1, -15, 352, 4316, 20448, 17264, 5632, -960, 256},

%e {1, -17, 638, 16500, 126968, 245872, 142752, 11200, 2432, -512},

%e {1, -19, 1184, 57472, 709232, 2490976, 2836928, 919552, 75776, -4864, 1024},

%e {1, -21, 2246, 190040, 3646816, 20950880, 41960896, 29090048, 6165760, 231168, 11776, -2048}

%t p[x_, n_] = (1 - 2x)^(n + 1)*(1 + Sum[2^(k - 1)*k^n*x^k, {k, 1, Infinity}]);

%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

%t Flatten[%]

%K sign,tabf,less

%O 0,4

%A _Roger L. Bagula_, May 16 2010

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)