OFFSET
1,3
COMMENTS
Row sums are:
{1, 3, 16, 285, 10938, 827935, 97511566, 15939477431, 3455244975656, 959962443311656,...}
This set of polynomials is a pure Infinite sum analogy to the Beta[n,m] integral.
Absolute values are used since the sums are zero otherwise.
FORMULA
p(x,n,m)=(1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, {k, 0, Infinity}]/4
EXAMPLE
{1},
{1, 2},
{1, 3, 12},
{3, 10, 48, 224},
{10, 42, 226, 1620, 9040},
{40, 245, 1530, 10024, 95904, 720192},
{245, 1365, 10892, 93096, 744528, 8855616, 87805824},
{1225, 11326, 87696, 799344, 8702064, 87478464, 1179952128, 14662445184},
{11326, 80094, 836556, 8401344, 88416672, 1166821632, 14621202720, 214725774528, 3224633430784},
{73626, 855162, 7965636, 92284896, 1140890112, 14497754256, 213099779232, 3217766464832, 51230717283840, 905285119960064}
MATHEMATICA
p[x_, n_, m_] = (1 - x)^(n + m + 1)*Sum[k^(n - 1)*(1 - k)^(m - 1)*x^k, { k, 0, Infinity}]/4;
Flatten[Table[Table[Apply[Plus, Abs[CoefficientList[ FullSimplify[ExpandAll[p[x, n, m]]], x]]], {m, 1, n}], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,uned
AUTHOR
Roger L. Bagula, Nov 20 2009
STATUS
approved