A389697 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (3*k+1) * binomial(3*n-3*k+1,n-2*k)/(3*n-3*k+1).

1, 1, 2, 8, 38, 192, 1014, 5539, 31051, 177645, 1033079, 6088992, 36293454, 218393703, 1324938698, 8095172229, 49767868166, 307644770467, 1911017591186, 11922651533079, 74676762339040, 469397564333139, 2960050537085895, 18721418271517348, 118728019920840586

Offset

0,3

Links

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

Formula

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

Mathematica

Table[Sum[(-1)^k*(3*k+1)*Binomial[3*n-3*k+1, n-2*k]/(3*n-3*k+1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 13 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (-1)^k*(3*k+1)*binomial(3*n-3*k+1, n-2*k)/(3*n-3*k+1));
(Magma) [&+[(-1)^k* (3*k+1)*Binomial(3*n-3*k+1, n-2*k)/(3*n-3*k+1): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Nov 13 2025

Crossrefs

Cf. A089354, A390517, A390518.

Cf. A001764, A109957.

Sequence in context: A002003 A123164 A264229 * A059423 A112109 A345178

Adjacent sequences: A389694 A389695 A389696 * A389698 A389699 A389700

Keyword

nonn

Author

Seiichi Manyama, Nov 08 2025

Status

approved