A390969 a(n) = (1/(2*n+1)) * Sum_{k=0..n} (k+1)^3 * (2*k+1) * binomial(3*n-k,n-k).

1, 9, 54, 307, 1743, 9996, 58026, 340791, 2022735, 12118665, 73207368, 445468884, 2728222084, 16804731336, 104041748994, 647113502847, 4041607835055, 25337241137355, 159385419715830, 1005757360156395, 6364732471157655, 40384029198218160, 256858123259214360

Offset

0,2

Links

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

Formula

G.f.: g^5 * (1 + 4*x*g^2 + x^2*g^4), where g = 1+x*g^3 is the g.f. of A001764.

a(n) = 6*(23*n^2+30*n+10)*(3*n+2)!/((2*n+5)!*n!). - Tani Akinari, Dec 23 2025

Mathematica

Table[Sum[(k+1)^3*(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)^3*(2*k+1)*binomial(3*n-k, n-k))/(2*n+1);
(Magma) [&+[(k+1)^3*(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
(Maxima) a(n):=6*(23*n^2+30*n+10)*(3*n+2)!/((2*n+5)!*n!); /* Tani Akinari, Dec 23 2025 */

Crossrefs

Cf. A001764, A006013, A390968, A390970.

Sequence in context: A001392 A188428 A243415 * A276602 A079764 A079761

Adjacent sequences: A390966 A390967 A390968 * A390970 A390971 A390972

Keyword

nonn

Author

Seiichi Manyama, Nov 25 2025

Status

approved