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A187543 Binomial convolutions of the central Lah numbers (A187535).

%I A187543 #19 Oct 06 2019 07:39:04
%S A187543 1,4,80,2832,144576,9660480,798468480,78670609920,9002061573120,
%T A187543 1173384611804160,171641216823552000,27843893955582566400,
%U A187543 4961007038613633638400,963075987422089673932800,202333751987206944654950400
%N A187543 Binomial convolutions of the central Lah numbers (A187535).
%H A187543 Vincenzo Librandi, <a href="/A187543/b187543.txt">Table of n, a(n) for n = 0..200</a>
%F A187543 a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..n), where L(n) is a central Lah number.
%F A187543 E.g.f.: (1/2 + 1/Pi*K(16x))^2, where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
%F A187543 Recurrence: (n-1)*n^2*(4*n^2-15*n+13)*a(n) = 4*(n-1)*(48*n^5-292*n^4+672*n^3-747*n^2+399*n-76)*a(n-1) - 32*(96*n^7-1000*n^6+4408*n^5-10628*n^4+15034*n^3-12312*n^2+5265*n-854)*a(n-2) + 1024*(2*n-5)^2*(4*n^2-7*n+2)*(n-2)^4*a(n-3). - _Vaclav Kotesovec_, Aug 10 2013
%F A187543 a(n) ~ n! * log(n) * 2^(4*n-1) / (Pi^2 * n) * (1 + (gamma + Pi + 4*log(2)) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 06 2019
%p A187543 a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
%p A187543 seq(add(binomial(n,k)*a(k)*a(n-k), k=0..n),n=0..12);
%t A187543 a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
%t A187543 Table[Sum[Binomial[n, k]a[k]a[n - k], {k, 0, n}], {n, 0, 20}]
%t A187543 CoefficientList[Series[(1/2 + EllipticK[16*x]/Pi)^2, {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Oct 06 2019 *)
%o A187543 (Maxima) a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
%o A187543 makelist(sum(binomial(n,k)*a(k)*a(n-k),k,0,n),n,0,12);
%Y A187543 Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187544, A187545, A187546, A187547, A187548.
%K A187543 nonn,easy
%O A187543 0,2
%A A187543 _Emanuele Munarini_, Mar 11 2011