A025902 Expansion of 1/((1-x^6)*(1-x^8)*(1-x^9)).

1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 3, 2, 2, 4, 1, 3, 3, 3, 3, 5, 2, 4, 4, 3, 3, 6, 3, 5, 5, 4, 4, 7, 3, 6, 6, 5, 5, 8, 4, 7, 7, 6, 6, 9, 5, 8, 8, 7, 7, 11, 6, 9, 9, 8, 8, 12, 7

Offset

0,19

Comments

a(n) is the number of partitions of n into parts 6, 8, and 9. - Michel Marcus, Jan 23 2024

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,0,1,1,0,0,0,0,-1,-1,0,-1,0,0,0,0,0,1).

Formula

a(n) = floor((n^2+7*n+373)/864 + (n+12)*(-1)^n/96 + (n+12)*((n+2) mod 3)/54 +(((n+4) mod 8)-((n+2) mod 8))/72). - Hoang Xuan Thanh, Sep 25 2025

Mathematica

CoefficientList[ Series[1/((1-x^6)*(1-x^8)*(1-x^9)), {x, 0, 100}], x] (* G. C. Greubel, Jan 23 2024 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^6)*(1-x^8)*(1-x^9)) )); // G. C. Greubel, Jan 23 2024
(SageMath)
def A025902_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^6)*(1-x^8)*(1-x^9))).list()
A025902_list(100) # G. C. Greubel, Jan 23 2024
(PARI) a(n) = ((n^2+7*n+289)/864 + n*(-1)^n/96 + n*((n+2)%3)/54 + (2/27)*[9, 0, 4, 4, 1, 1][n%6+1])\1 \\ Hoang Xuan Thanh, Sep 25 2025

Crossrefs

Cf. A025896, A025897, A025898, A025899, A025900, A025901, A025903.

Sequence in context: A214020 A029425 A219055 * A219923 A286950 A210638

Adjacent sequences: A025899 A025900 A025901 * A025903 A025904 A025905

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved