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

0,2

Links

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

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

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 *)

Prog

(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);

Crossrefs

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

Sequence in context: A012127 A189791 A057875 * A268734 A012824 A012057

Adjacent sequences: A187540 A187541 A187542 * A187544 A187545 A187546

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 11 2011

Status

approved