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A228902 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * y^k ), as read by rows.

%I A228902 #8 Feb 24 2025 02:09:00
%S A228902 1,1,1,1,3,1,1,6,45,1,1,10,505,2905,1,1,15,3045,412044,411500,1,1,21,
%T A228902 12880,16106168,1218805926,100545716,1,1,28,43176,309616264,
%U A228902 479536629727,9030648908720,37614371968,1,1,36,122640,3752248896,61545730104024,50139332516318674,139855355007409180,19977489354808,1
%N A228902 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * y^k ), as read by rows.
%e A228902 This triangle begins:
%e A228902   1;
%e A228902   1, 1;
%e A228902   1, 3, 1;
%e A228902   1, 6, 45, 1;
%e A228902   1, 10, 505, 2905, 1;
%e A228902   1, 15, 3045, 412044, 411500, 1;
%e A228902   1, 21, 12880, 16106168, 1218805926, 100545716, 1;
%e A228902   1, 28, 43176, 309616264, 479536629727, 9030648908720, 37614371968, 1;
%e A228902   1, 36, 122640, 3752248896, 61545730104024, 50139332516318674, 139855355007409180, 19977489354808, 1;
%e A228902   ...
%e A228902 G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+45*y^2+y^3)*x^3 + (1+10*y+505*y^2+2905*y^3+y^4)*x^4 + (1+15*y+3045*y^2+412044*y^3+411500*y^4+y^5)*x^5 +...
%e A228902 The logarithm of the g.f. equals the series:
%e A228902 log(A(x,y)) = (1 + y)*x
%e A228902    + (1 + 4*y + y^2)*x^2/2
%e A228902    + (1 + 9*y + 126*y^2 + y^3)*x^3/3
%e A228902    + (1 + 16*y + 1820*y^2 + 11440*y^3 + y^4)*x^4/4
%e A228902    + (1 + 25*y + 12650*y^2 + 2042975*y^3 + 2042975*y^4 + y^5)*x^5/5
%e A228902    + (1 + 36*y + 58905*y^2 + 94143280*y^3 + 7307872110*y^4 + 600805296*y^5 + y^6)*x^/6
%e A228902    + ...
%e A228902 in which the coefficients form A226234(n,k) = binomial(n^2, k^2).
%o A228902 (PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m^2, j^2)*y^j))+x*O(x^n)), n, x), k, y)}
%o A228902 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y A228902 Cf. A206848 (row sums), A206850 (antidiagonal sums), A228903 (diagonal).
%Y A228902 Cf. related triangles:  A226234 (log), A209196, A228900, A228904.
%K A228902 nonn,tabl
%O A228902 0,5
%A A228902 _Paul D. Hanna_, Sep 07 2013