A188676 Alternate partial sums of the binomial coefficients binomial(3*n,n).

1, 2, 13, 71, 424, 2579, 15985, 100295, 635176, 4051649, 25993366, 167543354, 1084134346, 7038291098, 45821937982, 299045487602, 1955803426045, 12815265660680, 84111082917925, 552872886403775, 3638971619401720

Offset: 0

Author

Emanuele Munarini, Apr 08 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A005809, A001764, A104859, A188678, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

Programs

Mathematica

Table[Sum[Binomial[3k, k](-1)^(n-k), {k, 0, n}], {n, 0, 20}]

Maxima

makelist(sum(binomial(3*k,k)*(-1)^(n-k),k,0,n),n,0,20);

Formula

a(n) = sum(k=0..n, (-1)^(n-k)*binomial(3*k,k) ).

Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(23*n^2+67*n+48)*a(n+1)-3*(3*n+4)*(3n+5)*a(n)=0.

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

a(n) ~ 3^(3*n+7/2)/(62*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012