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A218305 E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(x)*cosh(x).

%I A218305 #11 Apr 24 2024 08:55:25
%S A218305 1,1,8,148,4256,166816,8297600,500730112,35547379712,2902899914752,
%T A218305 268094176428032,27629598827044864,3143573312615481344,
%U A218305 391375817676973932544,52926434374336385122304,7725597721066205089890304,1210677595048894480252928000
%N A218305 E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(x)*cosh(x).
%C A218305 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218305 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%F A218305 E.g.f.: A(x) = Sum_{n>=0} (3*n+1)^(n-1) * cosh((3*n+1)*x) * x^n/n!.
%F A218305 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218305 E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/3 * LambertW(-3*x * exp(3*x)) ).
%F A218305 a(n) = 1/2 * Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k) for n > 0.
%F A218305 G.f.: 1/2 + 1/2 * Sum_{k>=0} (3*k+1)^(k-1) * x^k/(1 - (3*k+1)*x)^(k+1). (End)
%e A218305 E.g.f.: A(x) = 1 + x + 8*x^2/2! + 148*x^3/3! + 4256*x^4/4! + 166816*x^5/5! +...
%e A218305 where
%e A218305 A(x) = cosh(x) + 4^0*cosh(4*x)*x + 7^1*cosh(7*x)*x^2/2! + 10^2*cosh(10*x)*x^3/3! + 13^3*cosh(13*x)*x^4/4! + 16^4*cosh(16*x)*x^5/5! +...
%o A218305 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(3*X)*cosh(3*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
%o A218305 for(n=0,25,print1(a(n),", "))
%o A218305 (PARI) /* Formula derived from a LambertW identity: */
%o A218305 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(3*k+1)^(k-1)*cosh((3*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218305 for(n=0,25,print1(a(n),", "))
%Y A218305 Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218306, A218307, A218308, A218309, A218310.
%K A218305 nonn
%O A218305 0,3
%A A218305 _Paul D. Hanna_, Oct 25 2012