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URL: https://oeis.org/A066380

⇱ A066380 - OEIS

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A066380
a(n) = Sum_{k=0..n} binomial(3*n,k).
18
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
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k-1,k). - Seiichi Manyama, Jul 30 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 08 2025
From Seiichi Manyama, Aug 17 2025: (Start)
G.f.: 1/(1 - x*g^2*(6-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
G.f.: g/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)
D-finite with recurrence: -2160*(3*n + 2)*(3*n + 1)*a(n) + 18*(439*n^2 + 951*n + 542)*a(n + 1) - 12*(71*n^2 + 171*n + 32)*a(n + 2) - (11*n^2 + 213*n + 544)*a(n + 3) + 2*(n + 4)*(2*n + 7)*a(n + 4) = 0. - Robert Israel, Mar 13 2026
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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 23 2001
STATUS
approved