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 A247082 #33 Jan 16 2022 08:18:37
%S A247082 1,5,365,66605,22687565,12420052205,9972186170765,11039636939221805,
%T A247082 16116066766061589965,29996702068513925975405,
%U A247082 69334618695849722499185165,194843145588759580915489113005,654210085817395711127396030796365,2586566313303319454399746941903834605,11894287668430209899882926599828701863565
%N A247082 E.g.f.: (8 - 7*cosh(x)) / (13 - 12*cosh(x)).
%C A247082 The number of 3-level labeled linear rooted trees with 2*n leaves.
%C A247082 A bisection of A050351.
%C A247082 a(n) == 5 (mod 360) for n>0.
%H A247082 Seiichi Manyama, <a href="/A247082/b247082.txt">Table of n, a(n) for n = 0..195</a>
%F A247082 E.g.f.: 1/2 + (1/6)*Sum_{n>=0} exp(n^2*x) * (2/3)^n = Sum_{n>=0} a(n)*x^n/n!.
%F A247082 a(n) = Sum_{k=0..2*n} 2^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1. [After _Vladeta Jovovic_ in A050351]
%F A247082 a(n) ~ (2*n)! / (6 * (log(3/2))^(2*n+1)). - _Vaclav Kotesovec_, Nov 29 2014
%e A247082 E.g.f.: E(x) = 1 + 5*x^2/2! + 365*x^4/4! + 66605*x^6/6! + 22687565*x^8/8! +...
%e A247082 where E(x) = (8 - 7*cosh(x)) / (13 - 12*cosh(x)), or, equivalently,
%e A247082 E(x) = (7 - 16*exp(x) + 7*exp(2*x)) / (12 - 26*exp(x) + 12*exp(2*x)).
%e A247082 ALTERNATE GENERATING FUNCTION.
%e A247082 E.g.f.: A(x) = 1 + 5*x + 365*x^2/2! + 66605*x^3/3! + 22687565*x^4/4! +...
%e A247082 where
%e A247082 6*A(x) = 4 + exp(x)*(2/3) + exp(4*x)*(2/3)^2 + exp(9*x)*(2/3)^3 + exp(16*x)*(2/3)^4 + exp(25*x)*(2/3)^5 + exp(36*x)*(2/3)^6 + exp(49*x)*(2/3)^7 +...
%t A247082 nmax=20; Table[(CoefficientList[Series[(8-7*Cosh[x]) / (13-12*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]],{n,1,2*nmax+2,2}] (* _Vaclav Kotesovec_, Nov 29 2014 *)
%o A247082 (PARI) /* E.g.f.: (8 - 7*cosh(x)) / (13 - 12*cosh(x)): */
%o A247082 {a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (8 - 7*cosh(X)) / (13 - 12*cosh(X)) , 2*n)}
%o A247082 for(n=0, 20, print1(a(n), ", "))
%o A247082 (PARI) /* Formula for a(n): */
%o A247082 {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A247082 {a(n) = if(n==0, 1, sum(k=0, 2*n, 2^(k-1) * k! * Stirling2(2*n, k) ))}
%o A247082 for(n=0, 20, print1(a(n), ", "))
%o A247082 (PARI) /* As the Sum of an Infinite Series: */
%o A247082 \p60 \\ set precision
%o A247082 Vec(serlaplace(1/2+1/6*sum(n=0,2000,exp(n^2*x)*(2/3)^n*1.)))
%Y A247082 Cf. A249938, A249939, A249940, A250914, A250915, A050351.
%K A247082 nonn
%O A247082 0,2
%A A247082 _Paul D. Hanna_, Nov 28 2014