0,3

Robert Israel, Table of n, a(n) for n = 0..1498

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

G.f.: ( (1/x) * Series_Reversion( x * (1-x+x^2)^2 ) )^(1/2). - Seiichi Manyama, Mar 08 2025

D-finite with recurrence: -50*(5*n + 6)*(5*n + 2)*(5*n + 3)*(5*n + 4)*a(n) + 5*(3041*n^4 + 15872*n^3 + 32443*n^2 + 31300*n + 12096)*a(n + 1) - 4*(2*n + 5)*(n + 2)*(331*n^2 + 1025*n + 1044)*a(n + 2) + 36*(2*n + 5)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 3) = 0. - Robert Israel, Mar 08 2026

f:= gfun:-rectoproc({-50*(5*n + 6)*(5*n + 2)*(5*n + 3)*(5*n + 4)*a(n) + 5*(3041*n^4 + 15872*n^3 + 32443*n^2 + 31300*n + 12096)*a(n + 1) - 4*(2*n + 5)*(n + 2)*(331*n^2 + 1025*n + 1044)*a(n + 2) + 36*(2*n + 5)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 3), a(0) = 1, a(1) = 1, a(2) = 2}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Mar 08 2026

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

Cf. A000108, A137265, A200753, A200755.

Cf. A361245, A368969.

Sequence in context: A232207 A136563 A127077 * A104549 A174513 A000063

Adjacent sequences: A367024 A367025 A367026 * A367028 A367029 A367030

Seiichi Manyama, Nov 02 2023

approved