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A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ), as read by rows.

%I A209196 #16 Mar 30 2012 18:37:37
%S A209196 1,1,1,1,4,1,1,32,32,1,1,487,3282,487,1,1,11113,657573,657573,11113,1,
%T A209196 1,335745,209282906,1513844855,209282906,335745,1,1,12607257,
%U A209196 96673776804,5580284351032,5580284351032,96673776804,12607257,1,1,565877928,61162554558200
%N A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ), as read by rows.
%H A209196 Paul D. Hanna, <a href="/A209196/b209196.txt">Rows n = 0..30, flattened.</a>
%F A209196 Column 1 equals A209197.
%F A209196 Row sums equal A167006.
%F A209196 Antidiagonal sums equal A206830.
%e A209196 This triangle begins:
%e A209196 1;
%e A209196 1, 1;
%e A209196 1, 4, 1;
%e A209196 1, 32, 32, 1;
%e A209196 1, 487, 3282, 487, 1;
%e A209196 1, 11113, 657573, 657573, 11113, 1;
%e A209196 1, 335745, 209282906, 1513844855, 209282906, 335745, 1;
%e A209196 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1;
%e A209196 1, 565877928, 61162554558200, 31336815578461815, 229089181252258800, 31336815578461815, 61162554558200, 565877928, 1; ...
%e A209196 G.f.: A(x,y) = 1 + (1+y)*x + (1+4*y+y^2)*x^2 + (1+32*y+32*y^2+y^3)*x^3 + (1+487*y+3282*y^2+487*y^3+y^4)*x^4 +...
%e A209196 The logarithm of the g.f. equals the series:
%e A209196 log(A(x,y)) = (1 + y)*x
%e A209196 + (1 + 6*y + y^2)*x^2/2
%e A209196 + (1 + 84*y + 84*y^2 + y^3)*x^3/3
%e A209196 + (1 + 1820*y + 12870*y^2 + 1820*y^3 + y^4)*x^4/4
%e A209196 + (1 + 53130*y + 3268760*y^2 + 3268760*y^3 + 53130*y^4 + y^5)*x^5/5 +...
%e A209196 in which the coefficients form A209330(n,k) = binomial(n^2, n*k).
%o A209196 (PARI) {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,x^m/m*sum(j=0,m,binomial(m^2,m*j)*y^j))+x*O(x^n)),n,x),k,y)}
%o A209196 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A209196 Cf. A209197, A167006, A206830, A209330 (log), A155200.
%K A209196 nonn,tabl
%O A209196 0,5
%A A209196 _Paul D. Hanna_, Mar 05 2012