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

Author

Emanuele Munarini, Apr 08 2011

Keywords

• nonn
• easy

Links

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

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).

Programs

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 *)

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

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