A181145 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2n} C(2n,k)^2*y^k]*x^n/n ) = Sum_{n>=0,k=0..2n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.
1, 1, 4, 1, 1, 12, 27, 12, 1, 1, 24, 134, 236, 134, 24, 1, 1, 40, 410, 1540, 2380, 1540, 410, 40, 1, 1, 60, 975, 6260, 18386, 26216, 18386, 6260, 975, 60, 1, 1, 84, 1981, 19320, 91441, 227052, 306495, 227052, 91441, 19320, 1981, 84, 1, 1, 112, 3612, 49672
Offset: 0
Examples
G.f.: A(x,y) = 1 + (1+ 4*y+ y^2)*x + (1 + 12*y+ 27*y^2+ 12*y^3+ y^4)*x^2 + (1+ 24*y+ 134*y^2+ 236*y^3+ 134*y^4+ 24*y^5+ y^6)*x^3 +... The logarithm of the g.f. begins: log(A(x,y)) = (1 + 2^2*y + y^2)*x + (1 + 4^2*y + 6^2*y^2 + 4^2*y^3 + y^4)*x^2/2 + (1 + 6^2*y + 15^2*y^2 + 20^2*y^3 + 15^2*y^4 + 6^2*y^5 + y^6)*x^3/3 + (1 + 8^2*y + 28^2*y^2 + 56^2*y^3 + 70^2*y^4 + 56^2*y^5 + 28^2*y^6 + 8^2*y^7 + y^8)*x^4/4 +... Triangle begins: 1; 1, 4, 1; 1, 12, 27, 12, 1; 1, 24, 134, 236, 134, 24, 1; 1, 40, 410, 1540, 2380, 1540, 410, 40, 1; 1, 60, 975, 6260, 18386, 26216, 18386, 6260, 975, 60, 1; 1, 84, 1981, 19320, 91441, 227052, 306495, 227052, 91441, 19320, 1981, 84, 1; 1, 112, 3612, 49672, 344260, 1312080, 2883562, 3740572, 2883562, 1312080, 344260, 49672, 3612, 112, 1; ...
Crossrefs
Programs
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PARI
{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,2*m,binomial(2*m,j)^2*y^j)*x^m/m)+O(x^(n+1))),n,x),k,y)}
Formula
Row sums form A066357 (with offset), the number of ordered trees on 2n nodes with every subtree at the root having an even number of edges.
Extensions
Comment and example corrected by Paul D. Hanna, Oct 16 2010
Comments