A168217 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

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

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.

Links

Table of n, a(n) for n=1..39.

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

Sequence in context: A081323 A173958 A175243 * A329025 A317548 A320327

Adjacent sequences: A168214 A168215 A168216 * A168218 A168219 A168220

Keyword

nonn,uned

Author

Roger L. Bagula, Nov 20 2009

Status

approved