A385823 a(n) = Sum_{k=0..n} binomial(3*n-3,k).

1, 1, 7, 42, 256, 1586, 9949, 63004, 401930, 2579130, 16628809, 107636402, 699030226, 4552602248, 29722279084, 194458630304, 1274628824490, 8368726082346, 55027110808177, 362301656545966, 2388274575638228, 15760514137668514, 104108685843640517, 688331413734386356

Offset

0,3

Links

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

Formula

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

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

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-3,k) * binomial(3*n-k-4,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-4,n-k).

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

a(n) ~ 3^(3*n - 5/2) / (sqrt(Pi*n) * 2^(2*n-3)). - Vaclav Kotesovec, Oct 19 2025

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

D-finite with recurrence: 9504*(3*n - 1)*(3*n - 2)*a(n) + 36*(1259*n^2 + 4289*n + 2974)*a(n + 1) - 6*(4843*n^2 + 17134*n + 14696)*a(n + 2) + (4517*n^2 + 20285*n + 19872)*a(n + 3) - 110*(n + 4)*(2*n + 3)*a(n + 4) = 0. - Robert Israel, Mar 13 2026

Maple

f:= gfun:-rectoproc({9504*(3*n - 1)*(3*n - 2)*a(n) + 36*(1259*n^2 + 4289*n + 2974)*a(n + 1) - 6*(4843*n^2 + 17134*n + 14696)*a(n + 2) + (4517*n^2 + 20285*n + 19872)*a(n + 3) - 110*(n + 4)*(2*n + 3)*a(n + 4), a(0) = 1, a(1) = 1, a(2) = 7, a(3) = 42, a(4) = 256}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 13 2026

Mathematica

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

Prog

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

Crossrefs

Cf. A066380, A160906, A386006, A387007, A387008, A387033.

Cf. A001764, A047099, A165817.

Sequence in context: A003949 A252700 A033133 * A082035 A127016 A152239

Adjacent sequences: A385820 A385821 A385822 * A385824 A385825 A385826

Keyword

nonn

Author

Seiichi Manyama, Aug 13 2025

Status

approved