A221971 - OEIS

%I #15 Apr 27 2016 12:05:44

%S 1,0,1,0,1,1,0,0,4,1,0,0,3,11,1,0,0,0,27,26,1,0,0,0,17,148,57,1,0,0,0,

%T 0,278,646,120,1,0,0,0,0,155,2590,2481,247,1,0,0,0,0,0,4073,18304,

%U 8805,502,1,0,0,0,0,0,2073,58427,109699,29682,1013,1,0,0,0

%N G.f.: A(x,y) = Sum_{n>=0} n! * x^n*y^n * Product_{k=1..n} (1 + k*x) / (1 + k*x*y + k^2*x^2*y).

%H Paul D. Hanna, <a href="/A221971/b221971.txt">Rows n = 0..32, flattened.</a>

%F Row sums equal A208237.

%F Central terms equal A110501, the Genocchi numbers of first kind (unsigned).

%F Columns sums equal A005439, the Genocchi numbers of second kind.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 0, 4, 1;

%e 0, 0, 3, 11, 1;

%e 0, 0, 0, 27, 26, 1;

%e 0, 0, 0, 17, 148, 57, 1;

%e 0, 0, 0, 0, 278, 646, 120, 1;

%e 0, 0, 0, 0, 155, 2590, 2481, 247, 1;

%e 0, 0, 0, 0, 0, 4073, 18304, 8805, 502, 1;

%e 0, 0, 0, 0, 0, 2073, 58427, 109699, 29682, 1013, 1;

%e 0, 0, 0, 0, 0, 0, 80712, 614819, 590254, 96648, 2036, 1;

%e 0, 0, 0, 0, 0, 0, 38227, 1665829, 5340996, 2948040, 307255, 4083, 1; ...

%o (PARI) {T(n,k)=polcoeff(polcoeff(sum(m=0, n,m!*x^m*y^m*prod(k=1, m, (1+k*x)/(1+k*x*y+k^2*x^2*y +x*O(x^n)))),n,x),k,y)}

%o for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

%Y Cf. A208237 (row sums), A110501 (central terms), A005439 (column sums), A136126.

%K nonn,tabl

%O 0,9

%A _Paul D. Hanna_, Feb 01 2013