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A186737 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} ((1 + x*A(x))^k - 1).

%I A186737 #25 Nov 18 2019 05:35:39
%S A186737 1,1,3,14,82,563,4390,38273,370090,3951524,46436236,598606795,
%T A186737 8431113021,129134793211,2139840753676,38161937079832,728817179712481,
%U A186737 14837170081079164,320656534559643735,7329984468062242588,176662137501934534082,4476371962949911027001,118946746110478235868525
%N A186737 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} ((1 + x*A(x))^k - 1).
%C A186737 This is an example of Peter Bala's identity (cf. A158690):
%C A186737 Sum_{n>=0} Product_{k=1..n} (q^k - 1) = Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) at q = 1+x*A(x).
%H A186737 Vaclav Kotesovec, <a href="/A186737/b186737.txt">Table of n, a(n) for n = 0..115</a>
%H A186737 Hsien-Kuei Hwang, and Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], p. 33, 2019.
%F A186737 G.f. satisfies:
%F A186737 (1) A(x) = Sum_{n>=0} (1+x*A(x))^(-n^2) * Product_{k=1..n} ((1+x*A(x))^(2*k-1) - 1).
%F A186737 (2) A(x) = 1/(1 - (q-1)/(q - (q^2-1)/(q^2 - (q^3-1)/(q^3 - (q^4-1)/(q^4 - (q^5-1)/(q^5 -...)))))), a continued fraction, where q = 1+x*A(x).
%F A186737 a(n) ~ 6*sqrt(2) * exp(Pi^2/24) * 12^n * n! / Pi^(2*n+2). - _Vaclav Kotesovec_, May 05 2014
%e A186737 G.f.: A(X) = 1 + x + 3*x^2 + 14*x^3 + 82*x^4 + 563*x^5 + 4390*x^6 +...
%e A186737 where
%e A186737 A(x) = 1 + ((1+x*A(x))-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1)*((1+x*A(x))^3-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1)*((1+x*A(x))^3-1)*((1+x*A(x))^4-1) +...
%e A186737 Let q = 1+x*A(x), then:
%e A186737 A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...
%t A186737 nmax = 22; sol = {a[0] -> 1};
%t A186737 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Sum[Product[ ((1 + x A[x])^k - 1), {k, 1, j}], {j, 0, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A186737 sol /. Rule -> Set;
%t A186737 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)
%o A186737 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,prod(k=1,m,(1+x*A+x*O(x^n))^k - 1)));polcoeff(A,n)}
%o A186737 for(n=0,20,print1(a(n),", "))
%o A186737 (PARI) {a(n)=local(A=1+x, q); for(i=1,n,q=(1+x*A +x*O(x^n)); A=sum(m=0, n, q^(-m^2)*prod(k=1, m, (q^(2*k-1)-1)))); polcoeff(A, n)}
%o A186737 for(n=0, 20, print1(a(n), ", "))
%Y A186737 Cf. A179525.
%K A186737 nonn
%O A186737 0,3
%A A186737 _Paul D. Hanna_, Apr 23 2013