This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A218302 #13 Apr 23 2024 15:17:16
%S A218302 1,4,40,496,7488,134784,2836736,68635648,1881948160,57777184768,
%T A218302 1965962575872,73503311167488,2997314388623360,132455836580577280,
%U A218302 6308164435588415488,322185156718017642496,17571327124936467677184,1019377026461494381903872
%N A218302 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(4*x)*cosh(4*x).
%C A218302 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218302 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%F A218302 E.g.f.: A(x) = Sum_{n>=0} 4*(n+4)^(n-1) * cosh((n+4)*x) * x^n/n!.
%F A218302 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218302 E.g.f.: A(x) = 1/2 + 1/2 * exp( 4*x - 4*LambertW(-x * exp(x)) ).
%F A218302 a(n) = 2 * Sum_{k=0..n} (k+4)^(n-1) * binomial(n,k) for n > 0.
%F A218302 G.f.: 1/2 + 2 * Sum_{k>=0} (k+4)^(k-1) * x^k/(1 - (k+4)*x)^(k+1). (End)
%e A218302 E.g.f.: A(x) = 1 + 4*x + 40*x^2/2! + 496*x^3/3! + 7488*x^4/4! +...
%e A218302 where
%e A218302 A(x) = cosh(4*x) + 4*5^0*cosh(5*x)*x + 4*6^1*cosh(6*x)*x^2/2! + 4*7^2*cosh(7*x)*x^3/3! + 4*8^3*cosh(8*x)*x^4/4! + 4*9^4*cosh(9*x)*x^5/5! +...
%o A218302 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(4*R)*cosh(4*R); n!*polcoeff(Egf,n)}
%o A218302 for(n=0,25,print1(a(n),", "))
%o A218302 (PARI) /* Formula derived from a LambertW identity: */
%o A218302 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,4*(k+4)^(k-1)*cosh((k+4)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218302 for(n=0,25,print1(a(n),", "))
%Y A218302 Cf. A201595, A218300, A218301, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310.
%K A218302 nonn
%O A218302 0,2
%A A218302 _Paul D. Hanna_, Oct 25 2012