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

%I A187665 #11 May 30 2025 05:28:25
%S A187665 1,3,47,1440,67533,4280175,341307292,32750424588,3670267277749,
%T A187665 470237282353989,67781221867781615,10855095004543985756,
%U A187665 1912103925425230231884,367398970712627913234708,76469792506315229551855080
%N A187665 Binomial convolution of the central Lah numbers and the central Stirling numbers of the second kind.
%F A187665 a(n) = Sum_{k=0..n} binomial(n,k)*A187535(k)* A048993(2n-2k,n-k).
%F A187665 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
%p A187665 A048993 := proc(n,k) combinat[stirling2](n, k) ; end proc:
%p A187665 A187535 := proc(n) if n=0 then 1 else binomial(2*n-1, n-1)*(2*n)!/n! end if; end proc:
%p A187665 A187665 := proc(n) add(binomial(n,k)*A187535(k)*A048993(2*n-2*k,n-k), k=0..n) ; end proc:
%p A187665 seq(A187665(n),n=0..10)  ; # _R. J. Mathar_, Mar 28 2011
%t A187665 L[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
%t A187665 Table[Sum[Binomial[n,k]L[k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 14}]
%o A187665 (Maxima) L(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
%o A187665 makelist(sum(binomial(n,k)*L(k)*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);
%K A187665 nonn,easy
%O A187665 0,2
%A A187665 _Emanuele Munarini_, Mar 12 2011