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A215364 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*cosh(x*A(x)).

%I A215364 #21 Feb 16 2025 09:40:56
%S A215364 1,1,4,33,408,6725,139200,3475717,101722880,3416079753,129507425280,
%T A215364 5471712276041,254965505507328,12990483544072333,718474796305989632,
%U A215364 42871067358096134445,2745230569464318197760,187780115708775158008337,13665196427126843296972800
%N A215364 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*cosh(x*A(x)).
%F A215364 E.g.f. satisfies:
%F A215364 (1) A(x) = (1/x)*Series_Reversion(x-x^2*cosh(x)).
%F A215364 (2) A(x) = 1/(1 - x*A(x)*cosh(x*A(x))).
%F A215364 (3) A(x-x^2*cosh(x)) = 1/(1-x*cosh(x)).
%F A215364 a(n) = [x^n/n!] 1/(1 - x*cosh(x))^(n+1) / (n+1).
%F A215364 a(n) ~ n^(n-1) * s*sqrt(1/(6-2*s-r^2*s^2+r^2*s^3)) / (exp(n) * r^n), where r = 0.2278231894714399793... and s = 1.855593992316816009... are the roots of the equations r*s*(2*cosh(r*s) + r*s*sinh(r*s)) = 1, 1 + r*s^2*cosh(r*s) = s. - _Vaclav Kotesovec_, Jan 13 2014
%F A215364 a(n) = Sum_{k=0..n} k! * binomial(n+k+1,k)/(n+k+1) * A185951(n,k). - _Seiichi Manyama_, Feb 16 2025
%e A215364 E.g.f: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6725*x^5/5! +...
%t A215364 CoefficientList[1/x*InverseSeries[Series[x-x^2*Cosh[x], {x, 0, 21}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o A215364 (PARI) {a(n)=n!*polcoeff(1/(1 - x*cosh(x+x*O(x^n)))^(n+1)/(n+1),n)}
%o A215364 (PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
%o A215364 for(n=0, 25, print1(a(n), ", "))
%o A215364 (PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
%o A215364 a(n) = sum(k=0, n, k!*binomial(n+k+1,k)/(n+k+1)*a185951(n, k)); \\ _Seiichi Manyama_, Feb 16 2025
%Y A215364 Cf. A162654, A185951, A215363, A213644.
%K A215364 nonn
%O A215364 0,3
%A A215364 _Paul D. Hanna_, Aug 08 2012