A141947 A manufactured symmetrical triangular sequence of coefficients based on: t(n,m)=(Gamma[1 - m + n] Hypergeometric2F1Regularized[1, 1 + 2 m - n, 2 + m, -1])/Gamma[ -2 m + n]. The function is taken have backward and half forward.

0, 0, 1, 1, 0, 3, 3, 0, 1, 7, 7, 1, 0, 4, 15, 15, 4, 0, 1, 11, 31, 31, 11, 1, 0, 5, 26, 63, 63, 26, 5, 0, 1, 16, 57, 127, 127, 57, 16, 1, 0, 6, 42, 120, 255, 255, 120, 42, 6, 0, 1, 22, 99, 247, 511, 511, 247, 99, 22, 1, 0, 7, 64, 219, 502, 1023, 1023, 502, 219, 64, 7, 0

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

Keywords

• nonn
• uned

Comments

Row sums are:

{0, 2, 6, 16, 38, 86, 188, 402, 846, 1760, 3630}.

The odd n row are the most interesting.

The function was abstracted from the Mathematica generating function for

A052509 by taking out the powers of two:

t(n,m)=(n - m)!*(2^(-m + n)/Gamma[1 - m + n] - Hypergeometric2F1[1, 1 + 2 m - n, 2 + m, -1]/(Gamma[2 + m] Gamma[ -2 m + n])).

Examples

			{0, 0},
{1, 1},
{0, 3, 3, 0},
{1, 7, 7, 1},
{0, 4, 15, 15, 4, 0},
{1, 11, 31, 31, 11, 1},
{0, 5, 26, 63, 63, 26, 5, 0},
{1, 16, 57, 127, 127, 57, 16, 1},
{0, 6, 42, 120, 255, 255, 120, 42, 6, 0},
{1, 22, 99, 247, 511, 511, 247, 99, 22, 1},
{0, 7, 64, 219, 502, 1023, 1023, 502, 219, 64, 7, 0}
		

Crossrefs

Cf. A052509.

Programs

Mathematica

In[97]:= Table[Join[Table[(Gamma[1-m+n] Hypergeometric2F1Regularized[1,1+2 m-n,2+m,-1])/Gamma[ -2 m+n],{m,Floor[n/2],0,-1}],Table[(Gamma[1-m+n] Hypergeometric2F1Regularized[1,1+2 m-n,2+m,-1])/Gamma[ -2 m+n],{m,0,Floor[n/2]}]],{n,0,10}]; Flatten[%]

Formula

t(n,m)=(Gamma[1 - m + n] Hypergeometric2F1Regularized[1, 1 + 2 m - n, 2 + m, -1])/Gamma[ -2 m + n].