A066798 a(n) = Sum_{i=1..n} binomial(6*i,3*i).

20, 944, 49564, 2753720, 157871240, 9233006540, 547490880980, 32795094564080, 1979734520212192, 120244316085073616, 7339672750101339356, 449852213026938118560, 27666867082225970134160

Offset: 1

Author

Benoit Cloitre, Jan 18 2002

Keywords

• nonn

Links

• Harry J. Smith, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A006134, A066796, A066797, A066802.

Programs

Maple

s := RootOf((s+8)*s^3*x-s+1, s):
series( (1+8/s)^(3/2)*(s-4)*s^5/(3*(s^4+8*s^3-s+1)*(s^2+4*s-8)) - 1/(1-x), x=0, 30); # Mark van Hoeij, May 02 2013

Mathematica

Accumulate[Table[Binomial[6n,3n],{n,20}]] (* Harvey P. Dale, Apr 04 2020 *)

PARI

{ a=0; for (n=1, 100, write("b066798.txt", n, " ", a+=binomial(6*n, 3*n)) ) } \\ Harry J. Smith, Mar 28 2010

Formula

G.f.: (1+8/s)^(3/2)*(s-4)*s^5/(3*(s^4+8*s^3-s+1)*(s^2+4*s-8)) - 1/(1-x) where (s+8)*s^3*x-s+1 = 0. - Mark van Hoeij, May 02 2013

a(n) ~ sqrt(3) * 64^(n+1) / (189*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 07 2019

D-finite with recurrence n*(3*n-1)*(3*n-2)*a(n) +(-585*n^3+873*n^2-370*n+40)*a(n-1) +8*(6*n-5)*(6*n-1)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jan 11 2025