A390971 - OEIS

A390971

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

0, 1, 7, 41, 234, 1340, 7752, 45353, 268180, 1601145, 9641775, 58503324, 357377672, 2196211368, 13568830772, 84235073385, 525193567528, 3287329123715, 20649480324165, 130131277892385, 822511545547650, 5212970618578800, 33122169547346880, 210940544166493596

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

G.f.: x*g^6 * (1 + x*g^2), where g = 1+x*g^3 is the g.f. of A001764.

MATHEMATICA

Table[Sum[k^2*(2*k+1)*Binomial[3*n-k, n-k]/(2*n+1), {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Dec 05 2025 *)

PROG

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

(Magma) [&+[k^2*(2*k+1)*Binomial(3*n-k, n-k)/(2*n+1): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Dec 05 2025