A288470 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n,2*k).

1, 2, 14, 92, 646, 4652, 34124, 253528, 1901638, 14368844, 109208164, 833981128, 6394017436, 49185717752, 379438594136, 2934361958192, 22741538394694, 176582855512588, 1373431963785332, 10698376362421096, 83447762846703796, 651690159076273192, 5095051571420324264, 39874449115469939152, 312350761370734541596

Offset

0,2

Comments

Row sums of A155495.

a(n) is the constant term in the expansion of (-1 + (1 + x + 1/x)^2)^n. - Seiichi Manyama, Nov 21 2019

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1109

Formula

a(n) = hypergeom([-n,-n,1/2-n],[1/2,1],-1).

n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1).

G.f.: sqrt((1-2*x+sqrt(1-8*x))/(2*(1-7*x-8*x^2))).

a(n) ~ 8^n / sqrt(3*Pi*n). - Vaclav Kotesovec, Nov 27 2017

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Feb 13 2024

Maple

f:= gfun:-rectoproc({n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1), a(0)=1, a(1)=2, a(2)=14}, a(n), remember):
map(f, [$0..30]);

Mathematica

Table[Sum[Binomial[n, k] Binomial[2 n, 2 k], {k, 0, n}], {n, 0, 24}] (* Michael De Vlieger, Jun 09 2017 *)

Prog

(PARI) {a(n) = polcoef((-1+(1+x+1/x)^2)^n, 0)} \\ Seiichi Manyama, Nov 21 2019

Crossrefs

Cf. A094061, A155495, A218045.

Sequence in context: A282051 A020063 A323561 * A351857 A341395 A072148

Adjacent sequences: A288467 A288468 A288469 * A288471 A288472 A288473

Keyword

nonn

Author

Robert Israel, Jun 09 2017

Status

approved