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A218303 E.g.f. A(x) satisfies A( x/(exp(2*x)*cosh(2*x)) ) = exp(x)*cosh(x).

%I A218303 #18 Apr 24 2024 08:55:33
%S A218303 1,1,6,76,1480,39056,1303904,52716224,2504292480,136741146880,
%T A218303 8439125550592,580959483530240,44138582550333440,3668643339883089920,
%U A218303 331143571990522060800,32258185015683531587584,3373221864252806213435392,376881845889001869159759872
%N A218303 E.g.f. A(x) satisfies A( x/(exp(2*x)*cosh(2*x)) ) = exp(x)*cosh(x).
%C A218303 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218303 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%H A218303 Vaclav Kotesovec, <a href="/A218303/b218303.txt">Table of n, a(n) for n = 0..300</a>
%F A218303 E.g.f.: A(x) = Sum_{n>=0} (2*n+1)^(n-1) * cosh((2*n+1)*x) * x^n/n!.
%F A218303 a(n) ~ c * 2^n * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/(4*sqrt(LambertW(exp(-1)))) = 0.535672560704567808218663129282561449... . - _Vaclav Kotesovec_, Jul 13 2014, updated Jun 10 2019
%F A218303 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218303 E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/2 * LambertW(-2*x * exp(2*x)) ).
%F A218303 a(n) = 1/2 * Sum_{k=0..n} (2*k+1)^(n-1) * binomial(n,k) for n > 0.
%F A218303 G.f.: 1/2 + 1/2 * Sum_{k>=0} (2*k+1)^(k-1) * x^k/(1 - (2*k+1)*x)^(k+1). (End)
%e A218303 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 76*x^3/3! + 1480*x^4/4! + 39056*x^5/5! +...
%e A218303 where
%e A218303 A(x) = cosh(x) + 3^0*cosh(3*x)*x + 5^1*cosh(5*x)*x^2/2! + 7^2*cosh(7*x)*x^3/3! + 9^3*cosh(9*x)*x^4/4! + 11^4*cosh(11*x)*x^5/5! +...
%o A218303 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(2*X)*cosh(2*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
%o A218303 for(n=0,25,print1(a(n),", "))
%o A218303 (PARI) /* Formula derived from a LambertW identity: */
%o A218303 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(2*k+1)^(k-1)*cosh((2*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218303 for(n=0,25,print1(a(n),", "))
%Y A218303 Cf. A201595, A218300, A218301, A218302, A218304, A218305, A218306, A218307, A218308, A218309, A218310.
%K A218303 nonn
%O A218303 0,3
%A A218303 _Paul D. Hanna_, Oct 25 2012