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 A218304 #11 Apr 24 2024 08:55:29
%S A218304 1,3,30,468,10248,291888,10282464,432631104,21195292800,1186054914816,
%T A218304 74676568432128,5226914768016384,402722750814750720,
%U A218304 33876716756962652160,3089713688099323502592,303723970839738425622528,32015024916407062538256384
%N A218304 E.g.f. A(x) satisfies A( x/(exp(2*x)*cosh(2*x)) ) = exp(3*x)*cosh(3*x).
%C A218304 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218304 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%F A218304 E.g.f.: A(x) = Sum_{n>=0} 3*(2*n+3)^(n-1) * cosh((2*n+3)*x) * x^n/n!.
%F A218304 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218304 E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3/2 * LambertW(-2*x * exp(2*x)) ).
%F A218304 a(n) = 3/2 * Sum_{k=0..n} (2*k+3)^(n-1) * binomial(n,k) for n > 0.
%F A218304 G.f.: 1/2 + 3/2 * Sum_{k>=0} (2*k+3)^(k-1) * x^k/(1 - (2*k+3)*x)^(k+1). (End)
%e A218304 E.g.f.: A(x) = 1 + 3*x + 30*x^2/2! + 468*x^3/3! + 10248*x^4/4! + 291888*x^5/5! +...
%e A218304 where
%e A218304 A(x) = cosh(3*x) + 3*5^0*cosh(5*x)*x + 3*7^1*cosh(7*x)*x^2/2! + 3*9^2*cosh(9*x)*x^3/3! + 3*11^3*cosh(11*x)*x^4/4! + 3*13^4*cosh(13*x)*x^5/5! +...
%o A218304 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(2*X)*cosh(2*X)))); Egf=exp(3*R)*cosh(3*R); n!*polcoeff(Egf,n)}
%o A218304 for(n=0,25,print1(a(n),", "))
%o A218304 (PARI) /* Formula derived from a LambertW identity: */
%o A218304 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,3*(2*k+3)^(k-1)*cosh((2*k+3)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218304 for(n=0,25,print1(a(n),", "))
%Y A218304 Cf. A201595, A218300, A218301, A218302, A218303, A218305, A218306, A218307, A218308, A218309, A218310.
%K A218304 nonn
%O A218304 0,2
%A A218304 _Paul D. Hanna_, Oct 25 2012