A187545 Stirling transform (of the first kind) of the central Lah numbers (A187535).

1, 2, 38, 1312, 66408, 4442088, 369791064, 36848702784, 4277191653888, 566809715422464, 84441103242634176, 13970100487593468480, 2541362625439551554880, 504185908064687887996800, 108336183242510523080868480

Offset: 0

Author

Emanuele Munarini, Mar 11 2011

Keywords

• nonn
• easy
• nice

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187546, A187547, A187548.

Programs

Maple

lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(abs(combinat[stirling1](n,k))*lahc(k), k=0..n), n=0..20);

Mathematica

lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Abs[StirlingS1[n, k]]*lahc[k], {k, 0, n}], {n, 0, 20}]

Maxima

lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(abs(stirling1(n,k))*lahc(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 (signless) Stirling numbers of the first kind and L(n) are the central Lah numbers.

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

a(n) ~ n! / (2*Pi*n * (1 - exp(-1/16))^n). - Vaclav Kotesovec, Apr 10 2018