cp's OEIS Frontend

A352117 Expansion of e.g.f. 1/sqrt(2 - exp(2*x)).

%I A352117 #31 Nov 18 2023 08:26:51
%S A352117 1,1,5,37,377,4921,78365,1473277,31938737,784384561,21523937525,
%T A352117 652667322517,21672312694697,782133969325801,30481907097849485,
%U A352117 1275870745561131757,57083444567425884257,2718602143583362124641,137315150097164841942245
%N A352117 Expansion of e.g.f. 1/sqrt(2 - exp(2*x)).
%F A352117 a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling2(n,k).
%F A352117 a(n) ~ 2^n * n^n / (log(2)^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Mar 05 2022
%F A352117 Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x/(1 - 4*x/(1 - 3*x/(1 - 8*x/(1 - ... - (2*n-1)*x/(1 - 4*n*x/(1 - ... ))))))). Cf. A346982.  - _Peter Bala_, Aug 22 2023
%F A352117 For n > 0, a(n) = Sum_{k=1..n} a(n-k)*(1-k/n/2)*binomial(n,k)*2^k. - _Tani Akinari_, Sep 06 2023
%F A352117 a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-2)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 18 2023
%t A352117 m = 18; Range[0, m]! * CoefficientList[Series[(2 - Exp[2*x])^(-1/2), {x, 0, m}], x] (* _Amiram Eldar_, Mar 05 2022 *)
%o A352117 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(2-exp(2*x))))
%o A352117 (PARI) a(n) = sum(k=0, n, 2^(n-k)*prod(j=0, k-1, 2*j+1)*stirling(n, k, 2));
%o A352117 (Maxima) a[n]:=if n=0 then 1 else sum(a[n-k]*(1-k/n/2)*binomial(n,k)*2^k,k,1,n);
%o A352117 makelist(a[n],n,0,50); /* _Tani Akinari_, Sep 06 2023 */
%Y A352117 Cf. A000670, A097397, A216794, A352118, A352119, A305404, A346982 - A346985.
%K A352117 nonn
%O A352117 0,3
%A A352117 _Seiichi Manyama_, Mar 05 2022