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

Author

Paul D. Hanna, Oct 16 2010

Keywords

• nonn
• tabl

Comments

Compare g.f. to that of the triangle A034870:

* exp( Sum_{n>=1} [Sum_{k=0..2n} C(2n,k)*y^k]*x^n/n )

which consists of the even numbered rows of Pascal's triangle.

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

Cf. A066357 (row sums), A181146 (main diagonal).

Cf. variants: A181143, A181144, A001263, A034870.

Programs

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