A226312 Sum_{k=0..n} k*binomial(n,k)^2*binomial(2*k,k).

0, 2, 20, 186, 1704, 15510, 140676, 1273230, 11508048, 103919022, 937787100, 8458728630, 76269112200, 687496910490, 6195793616460, 55827244680930, 502959206683296, 4530723835554270, 40809306881317068, 367548287590324902, 3310080578306654520

Offset

0,2

Links

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

Bruno Haible, Combinatorial proof of a binomial identity, 1992.

Formula

a(n) ~ 3^(2*n+1/2)/(2*Pi). - Vaclav Kotesovec, Jun 10 2013

Recurrence: (n-2)*n*(n-1)*a(n) = (n-2)*(10*n^2-10*n+3)*a(n-1) - 9*(n-1)^3*a(n-2). - Vaclav Kotesovec, Jun 10 2013

G.f.: 2*x*((5+3*x)*(1-9*x)^2*hypergeom([2/3, 2/3],[1],-27*x*(1-x)^2/(1-9*x)^2)-4*(1-x)*(1+3*x)^3*hypergeom([5/3, 5/3],[2],-27*x*(1-x)^2/(1-9*x)^2))/(1-9*x)^(13/3). - Mark van Hoeij, Apr 11 2014

Maple

f:=n->add(k*binomial(n, k)^2*binomial(2*k, k), k=0..n);
[seq(f(n), n=0..40)];

Mathematica

Table[Sum[k*Binomial[n, k]^2*Binomial[2*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 10 2013 *)

Crossrefs

Sequence in context: A279462 A037566 A125857 * A171076 A287999 A348886

Adjacent sequences: A226309 A226310 A226311 * A226313 A226314 A226315

Keyword

nonn

Author

N. J. A. Sloane, Jun 08 2013

Status

approved