A187665 Binomial convolution of the central Lah numbers and the central Stirling numbers of the second kind.

1, 3, 47, 1440, 67533, 4280175, 341307292, 32750424588, 3670267277749, 470237282353989, 67781221867781615, 10855095004543985756, 1912103925425230231884, 367398970712627913234708, 76469792506315229551855080

Offset: 0

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Programs

Maple

A048993 := proc(n,k) combinat[stirling2](n, k) ; end proc:
A187535 := proc(n) if n=0 then 1 else binomial(2*n-1, n-1)*(2*n)!/n! end if; end proc:
A187665 := proc(n) add(binomial(n,k)*A187535(k)*A048993(2*n-2*k,n-k), k=0..n) ; end proc:
seq(A187665(n),n=0..10)  ; # R. J. Mathar, Mar 28 2011

Mathematica

L[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n,k]L[k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 14}]

Maxima

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

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*A187535(k)* A048993(2n-2k,n-k).

a(n) ~ c * 16^n * (n-1)!, where c = (1/Pi) * Sum_{k>=0} abs(Stirling2(2*k,k)) / (k! * 2^(4*k+1)) = 0.172113078600558193773... - Vaclav Kotesovec, Jul 05 2021, updated May 30 2025