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

%I A187544 #15 Oct 06 2019 07:02:59
%S A187544 1,2,38,1310,66254,4428782,368444078,36691056110,4256199137774,
%T A187544 563672814445742,83921091641375918,13875375391723852910,
%U A187544 2522552600160248918894,500141581330626431059502,107400097037199576065830958
%N A187544 Stirling transform (of the second kind) of the central Lah numbers (A187535).
%H A187544 Vaclav Kotesovec, <a href="/A187544/b187544.txt">Table of n, a(n) for n = 0..300</a>
%F A187544 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.
%F A187544 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).
%F A187544 a(n) ~ n! / (2*Pi*n * (log(17/16))^n). - _Vaclav Kotesovec_, Oct 06 2019
%p A187544 a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
%p A187544 seq(sum(combinat[stirling2](n,k)*a(k), k=0..n),n=0..12);
%t A187544 a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
%t A187544 Table[Sum[StirlingS2[n, k]a[k], {k, 0, n}], {n, 0, 20}]
%t A187544 CoefficientList[Series[1/2 + EllipticK[16*(E^x - 1)]/Pi, {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Oct 06 2019 *)
%o A187544 (Maxima) a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
%o A187544 makelist(sum(stirling2(n,k)*a(k),k,0,n),n,0,12);
%Y A187544 Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187545, A187546, A187547, A187548.
%K A187544 nonn,easy
%O A187544 0,2
%A A187544 _Emanuele Munarini_, Mar 11 2011