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%I #5 Sep 07 2013 19:43:02
%S 1,1,1,1,3,1,1,15,15,1,1,155,484,155,1,1,2685,36068,36068,2685,1,1,
%T 65517,5082340,15763254,5082340,65517,1,1,2063205,1179126560,
%U 13201421078,13201421078,1179126560,2063205,1,1,79715229,411708127954,19954261054442,61092286569334,19954261054442,411708127954,79715229,1
%N Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, (n-k)*k) * y^k ), as read by rows.
%e This triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 15, 15, 1;
%e 1, 155, 484, 155, 1;
%e 1, 2685, 36068, 36068, 2685, 1;
%e 1, 65517, 5082340, 15763254, 5082340, 65517, 1;
%e 1, 2063205, 1179126560, 13201421078, 13201421078, 1179126560, 2063205, 1;
%e 1, 79715229, 411708127954, 19954261054442, 61092286569334, 19954261054442, 411708127954, 79715229, 1;
%e ...
%e G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+15*y+15*y^2+y^3)*x^3 + (1+155*y+484*y^2+155*y^3+y^4)*x^4 + (1+2685*y+36068*y^2+36068*y^3+2685*y^4+y^5)*x^5 +...
%e The logarithm of the g.f. equals the series:
%e log(A(x,y)) = (1 + y)*x
%e + (1 + 4*y + y^2)*x^2/2
%e + (1 + 36*y + 36*y^2 + y^3)*x^3/3
%e + (1 + 560*y + 1820*y^2 + 560*y^3 + y^4)*x^4/4
%e + (1 + 12650*y + 177100*y^2 + 177100*y^3 + 12650*y^4 + y^5)*x^5/5
%e + (1 + 376992*y + 30260340*y^2 + 94143280*y^3 + 30260340*y^4 + 376992*y^5 + y^6)*x^6/6 +...
%e in which the coefficients form A228836(n,k) = binomial(n^2, (n-k)*k).
%o (PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m^2, (m-j)*j)*y^j))+x*O(x^n)), n, x), k, y)}
%o for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y Cf. A207135 (row sums), A207137 (antidiagonal sums), A228901 (column 1).
%Y Cf. related triangles: A228836 (log), A209196, A228902, A228904.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Sep 07 2013
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