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A218310 E.g.f. A(x) satisfies A( x/(exp(5*x)*cosh(5*x)) ) = exp(x)*cosh(x).

%I A218310 #12 Apr 24 2024 08:48:18
%S A218310 1,1,12,364,17248,1118816,92306432,9251542784,1091729307648,
%T A218310 148280571406336,22785577791987712,3908379504145178624,
%U A218310 740274425760340901888,153456630172316832628736,34557831428406144298647552,8401098284435734877893033984
%N A218310 E.g.f. A(x) satisfies A( x/(exp(5*x)*cosh(5*x)) ) = exp(x)*cosh(x).
%C A218310 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218310 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%F A218310 E.g.f.: A(x) = Sum_{n>=0} (5*n+1)^(n-1) * cosh((5*n+1)*x) * x^n/n!.
%F A218310 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218310 E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/5 * LambertW(-5*x * exp(5*x)) ).
%F A218310 a(n) = 1/2 * Sum_{k=0..n} (5*k+1)^(n-1) * binomial(n,k) for n > 0.
%F A218310 G.f.: 1/2 + 1/2 * Sum_{k>=0} (5*k+1)^(k-1) * x^k/(1 - (5*k+1)*x)^(k+1). (End)
%e A218310 E.g.f.: A(x) = 1 + x + 12*x^2/2! + 364*x^3/3! + 17248*x^4/4! + 1118816*x^5/5! +...
%e A218310 where
%e A218310 A(x) = cosh(x) + 6^0*cosh(6*x)*x + 11^1*cosh(11*x)*x^2/2! + 16^2*cosh(16*x)*x^3/3! + 21^3*cosh(21*x)*x^4/4! + 26^4*cosh(26*x)*x^5/5! +...
%o A218310 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(5*X)*cosh(5*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
%o A218310 for(n=0,25,print1(a(n),", "))
%o A218310 (PARI) /* Formula derived from a LambertW identity: */
%o A218310 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(5*k+1)^(k-1)*cosh((5*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218310 for(n=0,25,print1(a(n),", "))
%Y A218310 Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309.
%K A218310 nonn
%O A218310 0,3
%A A218310 _Paul D. Hanna_, Oct 25 2012