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 A218300 #33 Feb 16 2025 08:33:18
%S A218300 1,2,12,104,1216,18112,329600,7108096,177549312,5046554624,
%T A218300 160947232768,5694342479872,221410157133824,9387011838312448,
%U A218300 431051678297358336,21316106766591721472,1129526392342026649600,63855305138514241257472,3836490516381680506241024
%N A218300 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(2*x)*cosh(2*x).
%C A218300 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218300 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%H A218300 Vaclav Kotesovec, <a href="/A218300/b218300.txt">Table of n, a(n) for n = 0..300</a>
%H A218300 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A218300 E.g.f.: A(x) = Sum_{n>=0} 2*(n+2)^(n-1) * cosh((n+2)*x) * x^n/n!.
%F A218300 E.g.f.: A(x) = 1 + Sum_{n>=0} 2*(n+2)^(n-1) * sinh((n+2)*x) * x^n/n!.
%F A218300 a(n) ~ c * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/LambertW(exp(-1))^2 = 14.5815783688217906961670551786416446... . - _Vaclav Kotesovec_, Jul 13 2014, updated Jun 10 2019
%F A218300 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218300 E.g.f.: A(x) = 1/2 + 1/2 * exp( 2*x - 2*LambertW(-x * exp(x)) ).
%F A218300 a(n) = Sum_{k=0..n} (k+2)^(n-1) * binomial(n,k) for n > 0.
%F A218300 G.f.: 1/2 + Sum_{k>=0} (k+2)^(k-1) * x^k/(1 - (k+2)*x)^(k+1). (End)
%e A218300 E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 104*x^3/3! + 1216*x^4/4! + 18112*x^5/5! +...
%e A218300 where
%e A218300 A(x) = cosh(2*x) + 2*3^0*cosh(3*x)*x + 2*4^1*cosh(4*x)*x^2/2! + 2*5^2*cosh(5*x)*x^3/3! + 2*6^3*cosh(6*x)*x^4/4! + 2*7^4*cosh(7*x)*x^5/5! +...
%t A218300 nmin = 0; nmax = 18; sol = {a[0] -> 1}; nsol = Length[sol];
%t A218300 Do[A[x_] = Sum[a[k] x^k/k!, {k, 0, n}] /. sol; eq = CoefficientList[ A[x/(Exp[x] Cosh[x])] - Exp[2x] Cosh[2x] + O[x]^(n+1), x][[nsol+1;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol+1, nmax}];
%t A218300 a /@ Range[nmin, nmax] /. sol (* _Jean-François Alcover_, Nov 06 2019 *)
%o A218300 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(2*R)*cosh(2*R); n!*polcoeff(Egf,n)}
%o A218300 for(n=0,25,print1(a(n),", "))
%o A218300 (PARI) /* Formula derived from a LambertW identity: */
%o A218300 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,2*(k+2)^(k-1)*cosh((k+2)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218300 for(n=0,25,print1(a(n),", "))
%Y A218300 Cf. A201595, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310, A217900.
%Y A218300 Cf. A202357.
%K A218300 nonn
%O A218300 0,2
%A A218300 _Paul D. Hanna_, Oct 25 2012