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

1, 4, 80, 2832, 144576, 9660480, 798468480, 78670609920, 9002061573120, 1173384611804160, 171641216823552000, 27843893955582566400, 4961007038613633638400, 963075987422089673932800, 202333751987206944654950400

Offset: 0

Author

Emanuele Munarini, Mar 11 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187544, A187545, A187546, A187547, A187548.

Programs

Maple

a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(binomial(n,k)*a(k)*a(n-k), k=0..n),n=0..12);

Mathematica

a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n, k]a[k]a[n - k], {k, 0, n}], {n, 0, 20}]
CoefficientList[Series[(1/2 + EllipticK[16*x]/Pi)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 06 2019 *)

Maxima

a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(binomial(n,k)*a(k)*a(n-k),k,0,n),n,0,12);

Formula

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..n), where L(n) is a central Lah number.

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).

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

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