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A187546 Stirling transform (of the first kind, with signs) of the central Lah numbers (A187535).

%I A187546 #21 Apr 10 2018 10:19:21
%S A187546 1,2,34,1096,51984,3262488,254943384,23853046656,2600024557248,
%T A187546 323588157732096,45276442446814656,7035574740347812800,
%U A187546 1202158966644148296000,224022356544364922931840,45215509996613004825121920
%N A187546 Stirling transform (of the first kind, with signs) of the central Lah numbers (A187535).
%H A187546 Vincenzo Librandi, <a href="/A187546/b187546.txt">Table of n, a(n) for n = 0..200</a>
%F A187546 a(n) = sum((-1)^(n-k)*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.
%F A187546 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).
%F A187546 a(n) ~ n! / (2*Pi*n * (exp(1/16) - 1)^n). - _Vaclav Kotesovec_, Apr 10 2018
%p A187546 lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
%p A187546 seq(add(combinat[stirling1](n,k)*lahc(k), k=0..n), n=0..20);
%t A187546 lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
%t A187546 Table[Sum[StirlingS1[n, k]*lahc[k], {k, 0, n}], {n, 0, 20}]
%o A187546 (Maxima) lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
%o A187546 makelist(sum(stirling1(n,k)*lahc(k),k,0,n),n,0,12);
%Y A187546 Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187545, A187547, A187548.
%K A187546 nonn,easy
%O A187546 0,2
%A A187546 _Emanuele Munarini_, Mar 11 2011