A176667 A triangle of polynomial coefficients:p(x,n)=Sum[(k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^(x - n)

1, 2, 1, 4, 5, 1, 8, 18, 9, 1, 16, 54, 51, 14, 1, 32, 140, 220, 115, 20, 1, 64, 328, 750, 685, 225, 27, 1, 128, 784, 2044, 3080, 1785, 399, 35, 1, 256, 2096, 5068, 10220, 10465, 4088, 658, 44, 1, 512, 4704, 16776, 25284, 43806, 30681, 8484, 1026, 54, 1, 1024

Offset

0,2

Comments

Row sums are:A007582;

{1, 3, 10, 36, 136, 528, 2080, 8256, 32896, 131328, 524800,...}.

Links

Table of n, a(n) for n=0..55.

Formula

p(x,n)=Sum[(k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^(x - n);

t(n,m)=coefficients(p(x,n))

Example

{1},
{2, 1},
{4, 5, 1},
{8, 18, 9, 1},
{16, 54, 51, 14, 1},
{32, 140, 220, 115, 20, 1},
{64, 328, 750, 685, 225, 27, 1},
{128, 784, 2044, 3080, 1785, 399, 35, 1},
{256, 2096, 5068, 10220, 10465, 4088, 658, 44, 1},
{512, 4704, 16776, 25284, 43806, 30681, 8484, 1026, 54, 1},
{1024, 2496, 61920, 79980, 118020, 163569, 79905, 16290, 1530, 65, 1}

Mathematica

Clear[p, x, n]
p[x_, n_] = Sum[(k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^(x - n);
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A007582

Sequence in context: A124237 A123876 A114164 * A126182 A154342 A143494

Adjacent sequences: A176664 A176665 A176666 * A176668 A176669 A176670

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Apr 23 2010

Status

approved