A172395 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A000085(n)*x^n.

1, 1, 1, 0, 1, 0, 4, 0, 27, 0, 248, 0, 2830, 0, 38232, 0, 593859, 0, 10401712, 0, 202601898, 0, 4342263000, 0, 101551822350, 0, 2573779506192, 0, 70282204726396, 0, 2057490936366320, 0, 64291032462761955, 0, 2136017303903513184, 0

Offset: 0

Author

Paul D. Hanna, Feb 06 2010

Keywords

• nonn

Comments

The e.g.f. of A000085 is exp(x+x^2/2) = Sum_{n>=0} A000085(n)*x^n/n!, where A000085(n) is the number of self-inverse permutations on n letters.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^4 + 4*x^6 + 27*x^8 + 248*x^10 +...
where G(x) = A(x*G(x)) is the o.g.f. of A000085:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 76*x^6 + 232*x^7 +...
while the e.g.f. of A000085 is given by:
exp(x+x^2/2) = 1 + x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 26*x^5/5! +...
		

Links

• Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Crossrefs

Cf. A000085, A000699, A172394 (variant).

Programs

PARI

{a(n)=local(G=sum(m=0,n,m!*polcoeff(exp(x+x^2/2+x*O(x^m)),m)*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G),n)}

Formula

a(2n-2) = A000699(n), the number of irreducible diagrams with 2n nodes, for n>=1.

a(2n-1) = 0 for n>=2, with a(1)=1.