A391123 Expansion of g^2/(1 - x^3*g), where g = 1+x*g^3 is the g.f. of A001764.

1, 2, 7, 31, 146, 740, 3932, 21595, 121621, 698531, 4075738, 24090628, 143943104, 868008178, 5275779056, 32287126534, 198785908710, 1230417286319, 7652034348752, 47790934109113, 299623825839926, 1885007824069313, 11896542983930103, 75297656812364698

Offset

0,2

Links

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

Robert Israel, Recurrence of order 20

Formula

a(n) = Sum_{k=0..floor(n/3)} (k+2) * binomial(3*n-8*k+2,n-3*k)/(3*n-8*k+2).

D-finite with recurrence of order 20 (see link). - Robert Israel, Jun 11 2026

Mathematica

Table[Sum[ (k+2)*Binomial[3*n -8*k+2, n-3*k]/(3*n-8*k+2), {k, 0, Floor[n/3]}], {n, 0, 26}] (* Vincenzo Librandi, Nov 30 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (k+2)*binomial(3*n-8*k+2, n-3*k)/(3*n-8*k+2));
(Magma) [&+[(k+2)*Binomial(3*n-8*k+2, n-3*k)/(3*n-8*k+2): k in [0..Floor(n/3)]] : n in [0..40] ]; // Vincenzo Librandi, Nov 30 2025

Crossrefs

Cf. A391124, A391127, A391130.

Cf. A001764.

Sequence in context: A349769 A358963 A055836 * A391124 A076177 A391127

Adjacent sequences: A391120 A391121 A391122 * A391124 A391125 A391126

Keyword

nonn,changed

Author

Seiichi Manyama, Nov 29 2025

Status

approved