A187544 Stirling transform (of the second kind) of the central Lah numbers (A187535).

1, 2, 38, 1310, 66254, 4428782, 368444078, 36691056110, 4256199137774, 563672814445742, 83921091641375918, 13875375391723852910, 2522552600160248918894, 500141581330626431059502, 107400097037199576065830958

Offset: 0

Author

Emanuele Munarini, Mar 11 2011

Keywords

• nonn
• easy

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187545, A187546, A187547, A187548.

Programs

Maple

a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(combinat[stirling2](n,k)*a(k), k=0..n),n=0..12);

Mathematica

a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[StirlingS2[n, k]a[k], {k, 0, n}], {n, 0, 20}]
CoefficientList[Series[1/2 + EllipticK[16*(E^x - 1)]/Pi, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 06 2019 *)

Maxima

a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(stirling2(n,k)*a(k),k,0,n),n,0,12);

Formula

a(n) = sum(S(n,k)*L(k),k=0..n), where S(n,k) are the Stirling numbers of the second kind and L(n) are the central Lah numbers.

E.g.f.: 1/2 + 1/Pi*K(16(exp(x)-1)) where K(z) is the elliptic integral of the first kind (defined as in Mathematica).

a(n) ~ n! / (2*Pi*n * (log(17/16))^n). - Vaclav Kotesovec, Oct 06 2019