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%I #10 Jul 10 2021 07:14:52
%S 0,2,36,708,13800,275010,5546016,112928648,2316836304,47818490310,
%T 991809240180,20655287811972,431646101898336,9046910585978538,
%U 190096907348726280,4003238051096801040,84468326787457833888,1785362928126983779374,37794508394454024115212,801183789256961484235340
%N a(n) = Sum_{k=0..n} k*C(n,k)^3*C(n+k,k), where C := binomial.
%H C. Elsner, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulas for sums involving binomial coefficients</a>, Fib. Q., 43,1 (2005), 31-45.
%t Table[Sum[k*Binomial[n, k]^3*Binomial[n + k, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 10 2021 *)
%o (PARI) a(n) = sum(k=0, n, k*binomial(n, k)^3*binomial(n+k,k)); \\ _Michel Marcus_, Mar 10 2016
%Y Cf. A112035.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 28 2005