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

1,6

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])).

Links

Table of n, a(n) for n=1..72.

Formula

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

Example

{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}

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[%]

Crossrefs

Cf. A052509.

Sequence in context: A104548 A085707 A320253 * A216804 A010607 A338116

Adjacent sequences: A141944 A141945 A141946 * A141948 A141949 A141950

Keyword

nonn,uned

Author

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

Status

approved