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 A250915 #14 Jan 16 2022 08:18:33
%S A250915 1,9,2169,1306809,1469709369,2656472295609,7042235448544569,
%T A250915 25740278881968596409,124066865052334175027769,
%U A250915 762445058190042799428289209,5818666543923901596429593478969,53987940899344324456042542132654009,598504142090716188282023260396781018169
%N A250915 E.g.f.: (32 - 31*cosh(x)) / (41 - 40*cosh(x)).
%C A250915 The number of 5-level labeled linear rooted trees with 2*n leaves.
%C A250915 A bisection of A050353.
%C A250915 a(n) == 9 (mod 2160) for n>0.
%H A250915 Seiichi Manyama, <a href="/A250915/b250915.txt">Table of n, a(n) for n = 0..180</a>
%F A250915 E.g.f.: 3/4 + (1/20)*Sum_{n>=0} exp(n^2*x) * (4/5)^n = Sum_{n>=0} a(n)*x^n/n!.
%F A250915 a(n) = Sum_{k=0..2*n} 4^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1.
%F A250915 a(n) ~ (2*n)! / (20 * (log(5/4))^(2*n+1)). - _Vaclav Kotesovec_, Nov 29 2014
%e A250915 E.g.f.: E(x) = 1 + 9*x^2/2! + 2169*x^4/4! + 1306809*x^6/6! + 1469709369*x^8/8! +...
%e A250915 where E(x) = (32 - 31*cosh(x)) / (41 - 40*cosh(x)).
%e A250915 ALTERNATE GENERATING FUNCTION.
%e A250915 E.g.f.: A(x) = 1 + 9*x + 2169*x^2/2! + 1306809*x^3/3! + 1469709369*x^4/4! +...
%e A250915 where
%e A250915 20*A(x) = 16 + exp(x)*(4/5) + exp(4*x)*(4/5)^2 + exp(9*x)*(4/5)^3 + exp(16*x)*(4/5)^4 + exp(25*x)*(4/5)^5 + exp(36*x)*(4/5)^6 +...
%t A250915 nmax=20; Table[(CoefficientList[Series[(32-31*Cosh[x]) / (41-40*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]],{n,1,2*nmax+2,2}] (* _Vaclav Kotesovec_, Nov 29 2014 *)
%o A250915 (PARI) /* E.g.f.: (32 - 31*cosh(x)) / (41 - 40*cosh(x)): */
%o A250915 {a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (32 - 31*cosh(X)) / (41 - 40*cosh(X)) , 2*n)}
%o A250915 for(n=0, 20, print1(a(n), ", "))
%o A250915 (PARI) /* Formula for a(n): */
%o A250915 {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A250915 {a(n) = if(n==0, 1, sum(k=0, 2*n, 4^(k-1) * k! * Stirling2(2*n, k) ))}
%o A250915 for(n=0, 20, print1(a(n), ", "))
%o A250915 (PARI) /* As the Sum of an Infinite Series: */
%o A250915 \p100 \\ set precision
%o A250915 Vec(serlaplace(3/4 + 1/20*sum(n=0,3000,exp(n^2*x)*(4/5)^n*1.)))
%Y A250915 Cf. A249938, A249939, A247082, A250914, A050353.
%K A250915 nonn
%O A250915 0,2
%A A250915 _Paul D. Hanna_, Nov 28 2014