A188675 Partial sums of the binomial coefficients binomial(3*n,n) (A005809).

1, 4, 19, 103, 598, 3601, 22165, 138445, 873916, 5560741, 35605756, 229142476, 1480820176, 9603245620, 62463474700, 407330900284, 2662179813931, 17433248900656, 114359597479261, 751343566800961, 4943188072606456

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000(terms 1..100 from Vincenzo Librandi)

Formula

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

Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(31*n^2+95*n+72)*a(n+1)+3*(3*n+4)(3*n+5)*a(n)=0.

G.f.: 2*cos((1/3)*arcsin(3*sqrt(3*x)/2))/((1-x)*sqrt(4-27*x)).

a(n) ~ sqrt(3)*27^(n+1)/(46*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

Mathematica

Table[Sum[Binomial[3k, k], {k, 0, n}], {n, 0, 20}]
Accumulate[Table[Binomial[3n, n], {n, 0, 20}]] (* Nearly 300 times faster than the program above. *) (* Harvey P. Dale, Sep 14 2024 *)

Prog

(Maxima) makelist(sum(binomial(3*k, k), k, 0, n), n, 0, 20);
(PARI) for(n=0, 25, print1(sum(k=0, n, binomial(3*k, k)), ", ")) \\ G. C. Greubel, Jan 27 2017

Crossrefs

Cf. A001764, A005809, A104859, A188676, A188678 - A188687.

Cf. A263134: Sum_{k=0..n} binomial(3*k+1,k).

Cf. A087413: Sum_{k=0..n} binomial(3*k+2,k).

Sequence in context: A383771 A151382 A234958 * A199876 A225029 A078940

Adjacent sequences: A188672 A188673 A188674 * A188676 A188677 A188678

Keyword

nonn,easy

Author

Emanuele Munarini, Apr 08 2011

Status

approved