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 A218308 #9 Apr 24 2024 08:48:26
%S A218308 1,3,42,1116,44616,2394288,161719200,13187258304,1261037553792,
%T A218308 138415816348416,17155627044653568,2370099000682257408,
%U A218308 361171910376568571904,60185513513709805350912,10887989148395358662270976,2125192867898778619536457728
%N A218308 E.g.f. A(x) satisfies A( x/(exp(4*x)*cosh(4*x)) ) = exp(3*x)*cosh(3*x).
%C A218308 More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
%C A218308 then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
%F A218308 E.g.f.: A(x) = Sum_{n>=0} (4*n+1)^(n-1) * cosh((4*n+1)*x) * x^n/n!.
%F A218308 From _Seiichi Manyama_, Apr 23 2024: (Start)
%F A218308 E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3/4 * LambertW(-4*x * exp(4*x)) ).
%F A218308 a(n) = 3/2 * Sum_{k=0..n} (4*k+3)^(n-1) * binomial(n,k) for n > 0.
%F A218308 G.f.: 1/2 + 3/2 * Sum_{k>=0} (4*k+3)^(k-1) * x^k/(1 - (4*k+3)*x)^(k+1). (End)
%e A218308 E.g.f.: A(x) = 1 + 3*x + 42*x^2/2! + 1116*x^3/3! + 44616*x^4/4! + 2394288*x^5/5! +...
%e A218308 where
%e A218308 A(x) = cosh(3*x) + 3*7^0*cosh(7*x)*x + 3*11^1*cosh(11*x)*x^2/2! + 3*15^2*cosh(15*x)*x^3/3! + 3*19^3*cosh(19*x)*x^4/4! + 3*23^4*cosh(23*x)*x^5/5! +...
%o A218308 (PARI) {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(4*X)*cosh(4*X)))); Egf=exp(3*R)*cosh(3*R); n!*polcoeff(Egf,n)}
%o A218308 for(n=0,25,print1(a(n),", "))
%o A218308 (PARI) /* Formula derived from a LambertW identity: */
%o A218308 {a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,3*(4*k+3)^(k-1)*cosh((4*k+3)*X)*x^k/k!); n!*polcoeff(Egf,n)}
%o A218308 for(n=0,25,print1(a(n),", "))
%Y A218308 Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218309, A218310.
%K A218308 nonn
%O A218308 0,2
%A A218308 _Paul D. Hanna_, Oct 25 2012