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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.

%I A172395 #8 Dec 27 2015 04:01:53
%S A172395 1,1,1,0,1,0,4,0,27,0,248,0,2830,0,38232,0,593859,0,10401712,0,
%T A172395 202601898,0,4342263000,0,101551822350,0,2573779506192,0,
%U A172395 70282204726396,0,2057490936366320,0,64291032462761955,0,2136017303903513184,0
%N 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.
%C A172395 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 A172395 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 A172395 a(2n-2) = A000699(n), the number of irreducible diagrams with 2n nodes, for n>=1.
%F A172395 a(2n-1) = 0 for n>=2, with a(1)=1.
%e A172395 G.f.: A(x) = 1 + x + x^2 + x^4 + 4*x^6 + 27*x^8 + 248*x^10 +...
%e A172395 where G(x) = A(x*G(x)) is the o.g.f. of A000085:
%e A172395 G(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 76*x^6 + 232*x^7 +...
%e A172395 while the e.g.f. of A000085 is given by:
%e A172395 exp(x+x^2/2) = 1 + x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 26*x^5/5! +...
%o A172395 (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 A172395 Cf. A000085, A000699, A172394 (variant).
%K A172395 nonn
%O A172395 0,7
%A A172395 _Paul D. Hanna_, Feb 06 2010