A066380 a(n) = Sum_{k=0..n} binomial(3*n,k).

1, 4, 22, 130, 794, 4944, 31180, 198440, 1271626, 8192524, 53009102, 344212906, 2241812648, 14637774688, 95786202688, 628002401520, 4124304597834, 27126202533252, 178651732923346, 1178005033926998, 7776048412324714

Offset

0,2

References

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 425.

Links

Stefano Spezia, Table of n, a(n) for n = 0..1200 (first 151 terms from Harry J. Smith)

Formula

a(n) ~ C(3n, n)(2 - 4/n + O(1/n^2)).

G.f.: (1-g)/((3*g-1)*(2*g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

G.f.: x*(d/dx)log((F(x)-1)/(2-F(x))), where F(x) is g.f. of A001764. - Vladimir Kruchinin, Jun 13 2014

a(0)=1, a(n) = 8*a(n-1) - (5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!). - Tani Akinari, Sep 02 2014

a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n)). - Ilya Gutkovskiy, Oct 25 2017

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024

a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/2, -n/2], [1+n, 3/2+n], 1). - Stefano Spezia, Apr 09 2024

Maple

A066380:=n->add(binomial(3*n, k), k=0..n): seq(A066380(n), n=0..20); # Wesley Ivan Hurt, Sep 18 2014

Mathematica

Table[Sum[Binomial[3 n, k], {k, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, May 27 2013 *)
a[n_] := 8^n - (2*n)/(n+1)*Binomial[3*n, n]*Hypergeometric2F1[1, -2*n+1, n+2, -1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 02 2013 *)

Prog

(PARI) { for (n=0, 150, a=0; for (k=0, n, a+=binomial(3*n, k)); write("b066380.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010
(Maxima) a[0]:1$ a[n]:=8*a[n-1]-(5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!)$ makelist(a[n], n, 0, 200); /* Tani Akinari, Sep 02 2014 */

Crossrefs

Cf. A001764, A032443, A066381, A371739.

Sequence in context: A261399 A155862 A088536 * A180899 A007195 A356283

Adjacent sequences: A066377 A066378 A066379 * A066381 A066382 A066383

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 23 2001

Status

approved