A132310 a(n) = 3^n*Sum_{ k=0..n } binomial(2*k,k)/3^k.

1, 5, 21, 83, 319, 1209, 4551, 17085, 64125, 240995, 907741, 3428655, 12990121, 49370963, 188229489, 719805987, 2760498351, 10615101273, 40920439119, 158106581157, 612166272291, 2374756691313, 9228369037659, 35918537840577

Offset: 0

Author

Paul D. Hanna, Aug 18 2007

Keywords

• nonn

Comments

Simpler definition from N. J. A. Sloane, Jan 21 2009. Colin Mallows and I studied this sequence on Feb 21 1981 in connection with integration over a regular (solid) hexagon.

Hankel transform is A137717. - Paul Barry, Apr 26 2009

Examples

			a(1) = C(2,1)*(1/1 + 1/2 + 1/1) = 2*(5/2) = 5;
a(2) = C(4,2)*(1/1 + 2/4 + 3/6 + 2/4 + 1/1) = 6*(7/2) = 21;
a(3) = C(6,3)*(1/1 + 3/6 + 6/15 + 7/20 + 6/15 + 3/6 + 1/1) = 20*(83/20) = 83.
2*a(6) = sum(A182411(7,i), i=0..6) = 3432+858+572+572+728+1092+1848 = 9102 = 2*4551. - _Bruno Berselli_, May 02 2012
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-10*x+33*x^2-36*x^3], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

a(n)=binomial(2*n,n)*sum(k=0,2*n, polcoeff((1+x+x^2)^n,k)/binomial(2*n,k) )

PARI

a(n)=sum(k=0,2*n,polcoeff((1+x+x^2)^n,k) * k!*(2*n-k)! / (n!)^2 )

Formula

a(n) = C(2n,n) * sum_{k=0..2n} trinomial(n,k)/C(2n,k) where trinomial(n,k) = [x^k] (1 + x + x^2)^n, where [x^k] denotes "coefficient of x^k in ...".

G.f.: A(x) = 1/sqrt(1 - 10*x + 33*x^2 - 36*x^3).

a(n) = sum_{k=0..2n} trinomial(n,k) * k!*(2*n-k)! / (n!)^2.

2*a(n) = sum(A182411(n+1,i), i=0..n). - Bruno Berselli, May 02 2012

D-finite with recurrence: n*a(n) = (7*n-2)*a(n-1) - 6*(2*n-1)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 4^(n+1)/sqrt(Pi*n) . - Vaclav Kotesovec, Oct 20 2012