A172395 - OEIS

%I #8 Dec 27 2015 04:01:53

%S 1,1,1,0,1,0,4,0,27,0,248,0,2830,0,38232,0,593859,0,10401712,0,

%T 202601898,0,4342263000,0,101551822350,0,2573779506192,0,

%U 70282204726396,0,2057490936366320,0,64291032462761955,0,2136017303903513184,0

%N 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.

%C 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.

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

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

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

%e G.f.: A(x) = 1 + x + x^2 + x^4 + 4*x^6 + 27*x^8 + 248*x^10 +...

%e where G(x) = A(x*G(x)) is the o.g.f. of A000085:

%e G(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 76*x^6 + 232*x^7 +...

%e while the e.g.f. of A000085 is given by:

%e exp(x+x^2/2) = 1 + x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 26*x^5/5! +...

%o (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)}

%Y Cf. A000085, A000699, A172394 (variant).

%K nonn

%O 0,7

%A _Paul D. Hanna_, Feb 06 2010