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A302535 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).

%I A302535 #22 Jun 18 2019 04:47:00
%S A302535 1,1,6,61,846,14746,310016,7665141,218827766,7106293246,259169817316,
%T A302535 10497928495506,467768758203676,22739720141372196,1197560448125948596,
%U A302535 67910602688355999461,4125144974025630599846,267199960610924528490486,18382741943990196237909476,1338585578875261292134492646,102848696213697953204782043556
%N A302535 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).
%H A302535 Paul D. Hanna, <a href="/A302535/b302535.txt">Table of n, a(n) for n = 0..300</a>
%F A302535 G.f. A(x) satisfies:
%F A302535 (1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).
%F A302535 (2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A007696(n)*x^n, the o.g.f. of the quartic factorials.
%F A302535 (3) A(x) = 1 + x*A(x)^2 * (A(x) + 5*x*A'(x)) / (A(x) + x*A'(x)).
%F A302535 (4) A(x) = 1/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - 5*x*A(x)/(1 - 8*x*A(x)/(1 - 9*x*A(x)/(1 - 12*x*A(x)/(1 - 13*x*A(x)/(1 - ...)))))))), a continued fraction.
%F A302535 a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n - 1/4)). - _Vaclav Kotesovec_, Jun 18 2019
%e A302535 G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 846*x^4 + 14746*x^5 + 310016*x^6 + 7665141*x^7 + 218827766*x^8 + 7106293246*x^9 + 259169817316*x^10 + ...
%e A302535 such that
%e A302535 A(x) = 1 + x*A(x) + 5*x^2*A(x)^2 + 45*x^3*A(x)^3 + 585*x^4*A(x)^4 + 9945*x^5*A(x)^5 + 208845*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1) + ...
%o A302535 (PARI) /* Series Reversion of Quartic Factorials g.f.: */
%o A302535 {a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m*prod(k=1,m-1,4*k + 1))+x^2*O(x^n)), n)}
%o A302535 for(n=0, 30, print1(a(n), ", "))
%o A302535 (PARI) /* Differential Equation: */
%o A302535 {a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 5*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
%o A302535 for(n=0, 30, print1(a(n), ", "))
%o A302535 (PARI) /* Continued fraction: */
%o A302535 {a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(4*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
%o A302535 for(n=0, 30, print1(a(n), ", "))
%Y A302535 Cf. A007696, A088368, A301363, A302100, A302565.
%K A302535 nonn
%O A302535 0,3
%A A302535 _Paul D. Hanna_, Apr 09 2018