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.
1, 1, 1, 1, 3, 1, 1, 6, 45, 1, 1, 10, 505, 2905, 1, 1, 15, 3045, 412044, 411500, 1, 1, 21, 12880, 16106168, 1218805926, 100545716, 1, 1, 28, 43176, 309616264, 479536629727, 9030648908720, 37614371968, 1, 1, 36, 122640, 3752248896, 61545730104024, 50139332516318674, 139855355007409180, 19977489354808, 1
Offset: 0
Examples
This triangle begins: 1; 1, 1; 1, 3, 1; 1, 6, 45, 1; 1, 10, 505, 2905, 1; 1, 15, 3045, 412044, 411500, 1; 1, 21, 12880, 16106168, 1218805926, 100545716, 1; 1, 28, 43176, 309616264, 479536629727, 9030648908720, 37614371968, 1; 1, 36, 122640, 3752248896, 61545730104024, 50139332516318674, 139855355007409180, 19977489354808, 1; ... 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 +... The logarithm of the g.f. equals the series: log(A(x,y)) = (1 + y)*x + (1 + 4*y + y^2)*x^2/2 + (1 + 9*y + 126*y^2 + y^3)*x^3/3 + (1 + 16*y + 1820*y^2 + 11440*y^3 + y^4)*x^4/4 + (1 + 25*y + 12650*y^2 + 2042975*y^3 + 2042975*y^4 + y^5)*x^5/5 + (1 + 36*y + 58905*y^2 + 94143280*y^3 + 7307872110*y^4 + 600805296*y^5 + y^6)*x^/6 + ... in which the coefficients form A226234(n,k) = binomial(n^2, k^2).
Crossrefs
Programs
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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)} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))