A073075 Generating function satisfies A(x) = exp(2*A(x)*x + 2*A(x^3)*x^3/3 + 2*A(x^5)*x^5/5 + 2*A(x^7)*x^7/7 +...).

1, 2, 6, 22, 86, 358, 1554, 6950, 31822, 148434, 702802, 3369046, 16319050, 79749294, 392711090, 1946732854, 9706813790, 48651303118, 244972282734, 1238621756174, 6286144819506, 32011282859598, 163517409895602, 837631563577814, 4301996341244810

Offset: 0

Author

Paul D. Hanna, Aug 17 2002

Keywords

• easy
• nonn

Comments

Which kind of trees is counted by this sequence (see formulas)? - Joerg Arndt, Mar 04 2015

Programs

Maple

spec := [S,{B=Set(S),C=PowerSet(S),S=Prod(Z,B,C)},unlabeled]: seq(combstruct[count](spec,size=n), n=1..20); # Vladeta Jovovic, Feb 10 2005

Mathematica

m = 23; A[_] = 0;
Do[A[x_] = Exp[Sum[2 A[x^k] x^k/k, {k, 1, m, 2}]] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)

Formula

G.f.: A(x) = exp(sum_{n>=0} 2*A(x^(2n+1))*x^(2n+1)/(2n+1)), A(0)=1, where A(x) = 1 + 2x + 6x^2 + 22x^3 + 86x^4 + 358x^5 +...

Let b(n) = a(n-1) for n>=1, then sum(n>=1, b(n)*x^n ) = x * prod(n>=1, ((1+x^n)/(1-x^n))^b(n) ); compare to A000081, A004111, and A115593. - Joerg Arndt, Mar 04 2015