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A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * y^k ), as read by rows.

%I A228904 #5 Sep 07 2013 19:56:26
%S A228904 1,1,1,1,2,1,1,3,7,1,1,4,26,62,1,1,5,70,1087,1031,1,1,6,155,9257,
%T A228904 124702,24782,1,1,7,301,51397,4479983,26375325,774180,1,1,8,532,
%U A228904 215129,79666708,5059028293,8735721640,29763855,1,1,9,876,736410,891868573,357346615545,10783389596184,4162906254188,1359654560,1
%N A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * y^k ), as read by rows.
%e A228904 This triangle begins:
%e A228904 1;
%e A228904 1, 1;
%e A228904 1, 2, 1;
%e A228904 1, 3, 7, 1;
%e A228904 1, 4, 26, 62, 1;
%e A228904 1, 5, 70, 1087, 1031, 1;
%e A228904 1, 6, 155, 9257, 124702, 24782, 1;
%e A228904 1, 7, 301, 51397, 4479983, 26375325, 774180, 1;
%e A228904 1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1;
%e A228904 1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1;
%e A228904 ...
%e A228904 G.f.: A(x,y) = 1 + (1+y)*x + (1+2*y+y^2)*x^2 + (1+3*y+7*y^2+y^3)*x^3 + (1+4*y+26*y^2+62*y^3+y^4)*x^4 + (1+5*y+70*y^2+1087*y^3+1031*y^4+y^5)*x^5 +...
%e A228904 The logarithm of the g.f. equals the series:
%e A228904 log(A(x,y)) = (1 + y)*x
%e A228904 + (1 + 2*y + y^2)*x^2/2
%e A228904 + (1 + 3*y + 15*y^2 + y^3)*x^3/3
%e A228904 + (1 + 4*y + 70*y^2 + 220*y^3 + y^4)*x^4/4
%e A228904 + (1 + 5*y + 210*y^2 + 5005*y^3 + 4845*y^4 + y^5)*x^5/5
%e A228904 + (1 + 6*y + 495*y^2 + 48620*y^3 + 735471*y^4 + 142506*y^5 + y^6)*x^6/6 +...
%e A228904 in which the coefficients form A228832(n,k) = binomial(n*k, k^2).
%o A228904 (PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m*j, j^2)*y^j))+x*O(x^n)), n, x), k, y)}
%o A228904 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y A228904 Cf. A228809 (row sums), A228905 (antidiagonal sums), A228906 (diagonal).
%Y A228904 Cf. related triangles: A228832 (log), A209196, A228900, A228902.
%K A228904 nonn,tabl
%O A228904 0,5
%A A228904 _Paul D. Hanna_, Sep 07 2013