A390967 a(n) = (1/(n+1)) * Sum_{k=0..n} k^4 * (k+1) * binomial(2*n-k,n-k).

0, 1, 18, 134, 738, 3529, 15626, 66094, 271592, 1094970, 4357916, 17189832, 67381030, 262951065, 1022931210, 3970604310, 15388650480, 59579400030, 230518577340, 891563223876, 3447681609924, 13332176226474, 51561684003588, 199455373005004, 771769458006512

Offset

0,3

Links

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

Formula

G.f.: x*g^7 * (1 + 11*x*g + 11*x^2*g^2 + x^3*g^3), where g = 1+x*g^2 is the g.f. of A000108.

a(n) = 4*n*(154*n^3+81*n^2-19*n-6)*(2*n+1)!/((n+6)!*n!). - Tani Akinari, Dec 23 2025

Mathematica

Table[Sum[k^4*(k+1)*Binomial[2*n-k, n-k]/(n+1), {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Dec 05 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, k^4*(k+1)*binomial(2*n-k, n-k))/(n+1);
(Magma) [&+[k^4*(k+1)*Binomial(2*n-k, n-k)/(n+1): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Dec 05 2025
(Maxima) a(n):=4*n*(154*n^3+81*n^2-19*n-6)*(2*n+1)!/((n+6)!*n!); /* Tani Akinari, Dec 23 2025 */

Crossrefs

Cf. A390965, A390966.

Cf. A000108, A390964.

Sequence in context: A299368 A299932 A058649 * A273222 A103308 A320677

Adjacent sequences: A390964 A390965 A390966 * A390968 A390969 A390970

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 25 2025

Status

approved