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 A295239 #12 Oct 30 2024 11:41:45
%S A295239 1,-1,2,-9,68,-705,9234,-146209,2717000,-57986433,1397949830,
%T A295239 -37576332321,1114326129564,-36141571087297,1272713716466906,
%U A295239 -48360394499269665,1972269941821097744,-85929979225787811585,3983422470176606823054,-195765982110500512129057
%N A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4*x*exp(x))).
%F A295239 E.g.f.: 1/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + ...))))), a continued fraction.
%F A295239 a(n) ~ sqrt(2*(1+LambertW(-1/4))) * n^(n-1) / (exp(n) * (LambertW(-1/4))^n). - _Vaclav Kotesovec_, Nov 18 2017
%F A295239 a(n) = n! * Sum_{k=0..n} (-1)^k * k^(n-k) * A000108(k)/(n-k)!. - _Seiichi Manyama_, Oct 30 2024
%p A295239 a:=series(2/(1+sqrt(1+4*x*exp(x))),x=0,20): seq(n!*coeff(a,x,n),n=0..19); # _Paolo P. Lava_, Mar 27 2019
%t A295239 nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 + 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
%t A295239 nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t A295239 Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(-1)^m (m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 19}]
%o A295239 (PARI) a(n) = n!*sum(k=0, n, (-1)^k*k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ _Seiichi Manyama_, Oct 30 2024
%Y A295239 Cf. A000108, A006531, A052895, A295238.
%K A295239 sign
%O A295239 0,3
%A A295239 _Ilya Gutkovskiy_, Nov 18 2017