0,2

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

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

D-finite with recurrence: (54*n^2 + 216*n + 210)*a(n) + (829*n^2 + 6301*n + 10890)*a(n + 1) + (-3715*n^2 - 29341*n - 56640)*a(n + 2) + (6472*n^2 + 51418*n + 100506)*a(n + 3) + (-3715*n^2 - 29017*n - 50604)*a(n + 4) + (-822*n^2 - 15834*n - 65472)*a(n + 5) + (1048*n^2 + 15700*n + 58692)*a(n + 6) + (-236*n^2 - 3776*n - 15120)*a(n + 7) + (16*n^2 + 280*n + 1224)*a(n + 8) = 0. - Robert Israel, Dec 08 2025

f:= gfun:-rectoproc({(54*n^2 + 216*n + 210)*a(n) + (829*n^2 + 6301*n + 10890)*a(n + 1) + (-3715*n^2 - 29341*n - 56640)*a(n + 2) + (6472*n^2 + 51418*n + 100506)*a(n + 3) + (-3715*n^2 - 29017*n - 50604)*a(n + 4) + (-822*n^2 - 15834*n - 65472)*a(n + 5) + (1048*n^2 + 15700*n + 58692)*a(n + 6) + (-236*n^2 - 3776*n - 15120)*a(n + 7) + (16*n^2 + 280*n + 1224)*a(n + 8), a(0) = 1, a(1) = 2, a(2) = 6, a(3) = 25, a(4) = 119, a(5) = 606, a(6) = 3227, a(7) = 17751}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Dec 08 2025

Table[ Sum[(-1)^k*(3*k+2)*Binomial[3*n-3*k+2, n-2*k]/(3*n-3*k+2), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Dec 08 2025 *)

(PARI) a(n) = sum(k=0, n\2, (-1)^k*(3*k+2)*binomial(3*n-3*k+2, n-2*k)/(3*n-3*k+2));

(Magma) [&+[(-1)^k*(3*k+2)*Binomial(3*n-3*k+2, n-2*k)/(3*n-3*k+2): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 08 2025

Cf. A098746, A391295, A391297, A391301.

Cf. A001764, A391126.

Sequence in context: A006965 A202705 A058801 * A321720 A358499 A387498

Adjacent sequences: A391296 A391297 A391298 * A391300 A391301 A391302

Seiichi Manyama, Dec 06 2025

approved