A391064 Expansion of g/(1 - x^3*g^6), where g = 1+x*g^3 is the g.f. of A001764.

1, 1, 3, 13, 62, 315, 1674, 9193, 51759, 297174, 1733322, 10241823, 61176883, 368807090, 2241052623, 13711815645, 84403480214, 522329376663, 3247830666870, 20281159710438, 127133515076886, 799720690493010, 5046518811249591, 31937618370152547, 202659338107723146

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..floor(n/3)} (6*k+1) * binomial(3*n-3*k+1,n-3*k)/(3*n-3*k+1).

a(n) = (1/(2*n+1)) * Sum_{k=0..floor(n/3)} (6*k+1) * binomial(3*n-3*k,n-3*k).

Mathematica

Table[Sum[ (6*k+1)*Binomial[3*n -3*k+1, n-3*k]/(3*n-3*k+1), {k, 0, Floor[n/3]}], {n, 0, 26}] (* Vincenzo Librandi, Dec 01 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (6*k+1)*binomial(3*n-3*k+1, n-3*k)/(3*n-3*k+1));
(Magma) [&+[(6*k+1)*Binomial(3*n-3*k+1, n-3*k)/(3*n-3*k+1): k in [0..Floor(n/3)]] : n in [0..40] ]; // Vincenzo Librandi, Dec 01 2025

Crossrefs

Cf. A390485, A391059, A391061, A391062, A391063, A391065.

Cf. A001764, A047099.

Sequence in context: A122122 A093424 A389403 * A186242 A367060 A350478

Adjacent sequences: A391061 A391062 A391063 * A391065 A391066 A391067

Keyword

nonn

Author

Seiichi Manyama, Nov 26 2025

Status

approved