A390968 - OEIS

A390968

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

1, 5, 24, 121, 637, 3468, 19380, 110561, 641355, 3771885, 22439040, 134796060, 816540124, 4982228488, 30593078076, 188908910769, 1172307070375, 7307391855855, 45732182412840, 287246023045905, 1810153271613645, 11441454994023600, 72517477266539760

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OFFSET

0,2

LINKS

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

FORMULA

G.f.: g^4 * (1 + x*g^2), where g = 1+x*g^3 is the g.f. of A001764.

a(n) = 2*A006629(n) - A001764(n+1) = (3*n+2) * (3*n+3)!/((n+2)! * (2*n+3)!).

MATHEMATICA

Table[Sum[(k+1)^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+1)^2*(2*k+1)*binomial(3*n-k, n-k))/(2*n+1);

(Magma) [&+[(k+1)^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

CROSSREFS

Cf. A001764, A006013, A390969, A390970.

Cf. A006629, A070857, A000139, A013698.

Sequence in context: A017977 A017978 A291245 * A255715 A163610 A055825

Adjacent sequences: A390965 A390966 A390967 * A390969 A390970 A390971

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Nov 25 2025

STATUS

approved