A372160 E.g.f. A(x) satisfies A(x) = exp( 2 * x / (1 - x * A(x)^(1/2)) ).

1, 2, 8, 56, 568, 7592, 126364, 2522060, 58760272, 1566368432, 47036927284, 1571615915828, 57841636573912, 2325362549256008, 101399801919677356, 4767244262108645948, 240395075369097851296, 12943276401835227578720, 741127491503124866498404

Offset: 0

Author

Seiichi Manyama, Apr 20 2024

Keywords

• nonn
• changed

Crossrefs

Cf. A161630.

Programs

Mathematica

terms=19; A[]=1; Do[A[x] = Exp[2*x/(1-x*Sqrt[A[x]])] + O[x]^terms // Normal, terms];CoefficientList[Series[A[x],{x,0,terms}],x]Range[0,terms-1]! (* Stefano Spezia, Aug 26 2025 *)

PARI

a(n, r=2, s=1, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(n+(s-1)*k-1, n-k)/k!);

Formula

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A161630.

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.

a(n) ~ (1 + 2*LambertW(1/2))^(n + 1/2) * n^(n-1) / (sqrt(1 + LambertW(1/2)) * 2^(2*n+3) * exp(n) * LambertW(1/2)^(2*n + 7/2)). - Vaclav Kotesovec, Aug 27 2025