A187666 Binomial convolution of the (signless) central Lah numbers (A187535) and the (signless) central Stirling numbers of the first kind (A187646).

1, 3, 51, 1599, 74545, 4654255, 365549495, 34642467783, 3846064986001, 489429448820811, 70208261310969435, 11205444535728231855, 1969021774778391995761, 377672618542009829524551, 78507169034687468202172591

Offset: 0

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Crossrefs

Cf. A187535, A187646.

Programs

Maple

L := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(binomial(n,k)*L(k)*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);

Mathematica

L[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n,k]L[k]Abs[StirlingS1[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)*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * Lah(2k,k) * Stirling1(2n-2k,n-k).

a(n) ~ c * 2^(4*n + 1/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)), where c = Sum_{k>=0} abs(Stirling1(2*k,k)) / (k! * 2^(4*k+1)) = 0.550990257867992515027936630097897... - Vaclav Kotesovec, May 30 2025