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%I #2 Mar 30 2012 17:34:35
%S 1,1,2,1,3,12,3,10,48,224,10,42,226,1620,9040,40,245,1530,10024,95904,
%T 720192,245,1365,10892,93096,744528,8855616,87805824,1225,11326,87696,
%U 799344,8702064,87478464,1179952128,14662445184,11326,80094,836556
%N A triangular sequence based on the first level sum of polynomial coefficients: p(x,n,m)=(1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, {k, 0, Infinity}]/4
%C Row sums are:
%C {1, 3, 16, 285, 10938, 827935, 97511566, 15939477431, 3455244975656, 959962443311656,...}
%C This set of polynomials is a pure Infinite sum analogy to the Beta[n,m] integral.
%C Absolute values are used since the sums are zero otherwise.
%F p(x,n,m)=(1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, {k, 0, Infinity}]/4
%e {1},
%e {1, 2},
%e {1, 3, 12},
%e {3, 10, 48, 224},
%e {10, 42, 226, 1620, 9040},
%e {40, 245, 1530, 10024, 95904, 720192},
%e {245, 1365, 10892, 93096, 744528, 8855616, 87805824},
%e {1225, 11326, 87696, 799344, 8702064, 87478464, 1179952128, 14662445184},
%e {11326, 80094, 836556, 8401344, 88416672, 1166821632, 14621202720, 214725774528, 3224633430784},
%e {73626, 855162, 7965636, 92284896, 1140890112, 14497754256, 213099779232, 3217766464832, 51230717283840, 905285119960064}
%t p[x_, n_, m_] = (1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, { k, 0, Infinity}]/4;
%t Flatten[Table[Table[Apply[Plus, Abs[CoefficientList[ FullSimplify[ExpandAll[p[x, n, m]]], x]]], {m, 1, n}], {n, 1, 10}]]
%K nonn,uned
%O 1,3
%A _Roger L. Bagula_, Nov 20 2009
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