A391101 Expansion of g/(1 - x^3*g^4), where g = 1+x*g^4 is the g.f. of A002293.

1, 1, 4, 23, 145, 999, 7285, 55254, 431430, 3444731, 27995735, 230823083, 1925961450, 16232319212, 137987886985, 1181739515256, 10186216847417, 88303537214398, 769374846585827, 6733766490642597, 59175167640471740, 521930061845257745, 4618808781083220467

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..floor(n/3)} (4*k+1) * binomial(4*n-8*k+1,n-3*k)/(4*n-8*k+1).

Mathematica

Table[Sum[ (4*k+1)*Binomial[4* n-8*k+1, n-3*k]/(4*n-8*k+1), {k, 0, Floor[n/3]}], {n, 0, 26}] (* Vincenzo Librandi, Nov 30 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (4*k+1)*binomial(4*n-8*k+1, n-3*k)/(4*n-8*k+1));
(Magma) [&+[(4*k+1)*Binomial(4*n-8*k+1, n-3*k)/(4*n-8*k+1): k in [0..Floor(n/3)]] : n in [0..40] ]; // Vincenzo Librandi, Nov 30 2025

Crossrefs

Cf. A390298, A391098, A391100.

Cf. A002293, A391083.

Sequence in context: A020079 A391100 A367043 * A392516 A357152 A146964

Adjacent sequences: A391098 A391099 A391100 * A391102 A391103 A391104

Keyword

nonn

Author

Seiichi Manyama, Nov 28 2025

Status

approved