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%I #2 Oct 12 2012 14:54:57
%S 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,
%T 625,2012,1859,470,27,1,72,1708,9784,17190,9784,1708,72,1,1,152,5628,
%U 49384,134870,127464,41308,3992,81,1,232,14189,199616,884498,1431728,884498
%N 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}]]
%C Row sums are factorial:
%C {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800,...}.
%F 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}],
%F (1 - x)^(n + 1)*Sum[(1 + k + k^2)^Floor[n/2]*x^ k, {k, 0, Infinity}]]
%e {1},
%e {1},
%e {1, 0, 1},
%e {1, 2, 3},
%e {1, 4, 14, 4, 1},
%e {1, 12, 54, 44, 9},
%e {1, 20, 175, 328, 175, 20, 1},
%e {1, 46, 625, 2012, 1859, 470, 27},
%e {1, 72, 1708, 9784, 17190, 9784, 1708, 72, 1},
%e {1, 152, 5628, 49384, 134870, 127464, 41308, 3992, 81},
%e {1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1}
%t 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}],
%t (1 - x)^(n + 1)*Sum[(1 + k + k^2)^Floor[n/2]*x^k, {k, 0, Infinity}]]
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]
%t Flatten[%]
%K nonn,uned
%O 0,7
%A _Roger L. Bagula_ and _Gary W. Adamson_, Dec 03 2009