A141947 - OEIS

%I #2 Oct 12 2012 14:54:51

%S 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,

%T 26,5,0,1,16,57,127,127,57,16,1,0,6,42,120,255,255,120,42,6,0,1,22,99,

%U 247,511,511,247,99,22,1,0,7,64,219,502,1023,1023,502,219,64,7,0

%N 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.

%C Row sums are:

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

%C The odd n row are the most interesting.

%C The function was abstracted from the Mathematica generating function for

%C A052509 by taking out the powers of two:

%C 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])).

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

%e {0, 0},

%e {1, 1},

%e {0, 3, 3, 0},

%e {1, 7, 7, 1},

%e {0, 4, 15, 15, 4, 0},

%e {1, 11, 31, 31, 11, 1},

%e {0, 5, 26, 63, 63, 26, 5, 0},

%e {1, 16, 57, 127, 127, 57, 16, 1},

%e {0, 6, 42, 120, 255, 255, 120, 42, 6, 0},

%e {1, 22, 99, 247, 511, 511, 247, 99, 22, 1},

%e {0, 7, 64, 219, 502, 1023, 1023, 502, 219, 64, 7, 0}

%t 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[%]

%Y Cf. A052509.

%K nonn,uned

%O 1,6

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 14 2008