A386006 a(n) = Sum_{k=0..n} binomial(3*n-2,k).

1, 2, 11, 64, 386, 2380, 14893, 94184, 600370, 3850756, 24821333, 160645504, 1043243132, 6794414896, 44360053772, 290244832992, 1902631226010, 12493030680180, 82153313341429, 540953389469312, 3566279609565226, 23536562549993228, 155489358646406149

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] (1+x)^(3*n-2)/(1-x).

a(n) = [x^n] 1/((1-x)^(2*n-2) * (1-2*x)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-2,k) * binomial(3*n-k-3,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-3,n-k).

G.f.: 1/(g * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.

a(n) ~ 3^(3*n - 3/2) / (sqrt(Pi*n) * 2^(2*n-2)). - Vaclav Kotesovec, Oct 19 2025

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-1,n-2*k). - Seiichi Manyama, Nov 11 2025

D-finite with recurrence: 1968*(3*n - 1)*(3*n + 1)*a(n) - 2*(3067*n^2 + 2600*n - 1083)*a(n + 1) + 2*(233*n^2 + 13*n - 996)*a(n + 2) + (35*n^2 + 286*n + 539)*a(n + 3) - 2*(n + 4)*(2*n + 5)*a(n + 4) = 0. - Robert Israel, Mar 13 2026

Maple

f:= gfun:-rectoproc({1968*(3*n - 1)*(3*n + 1)*a(n) - 2*(3067*n^2 + 2600*n - 1083)*a(n + 1) + 2*(233*n^2 + 13*n - 996)*a(n + 2) + (35*n^2 + 286*n + 539)*a(n + 3) - 2*(n + 4)*(2*n + 5)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 11, a(3) = 64, a(4) = 386}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 13 2026

Mathematica

Table[Sum[Binomial[3*n-2, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 27 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(3*n-2, k));
(Magma) [&+[Binomial(3*n-2, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Crossrefs

Cf. A066380, A160906, A385823, A387007, A387008, A387033.

Cf. A001764, A047099, A165817.

Sequence in context: A179120 A373890 A038725 * A161947 A382918 A349023

Adjacent sequences: A386003 A386004 A386005 * A386007 A386008 A386009

Keyword

nonn

Author

Seiichi Manyama, Aug 13 2025

Status

approved