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

1, 1, 5, 37, 377, 4921, 78365, 1473277, 31938737, 784384561, 21523937525, 652667322517, 21672312694697, 782133969325801, 30481907097849485, 1275870745561131757, 57083444567425884257, 2718602143583362124641, 137315150097164841942245

Offset: 0

Author

Seiichi Manyama, Mar 05 2022

Keywords

• nonn

Crossrefs

Cf. A000670, A097397, A216794, A352118, A352119, A305404, A346982 - A346985.

Programs

Mathematica

m = 18; Range[0, m]! * CoefficientList[Series[(2 - Exp[2*x])^(-1/2), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

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);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 06 2023 */

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(2-exp(2*x))))

PARI

a(n) = sum(k=0, n, 2^(n-k)*prod(j=0, k-1, 2*j+1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling2(n,k).

a(n) ~ 2^n * n^n / (log(2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Mar 05 2022

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

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

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