A025916 Expansion of 1/((1-x^7)*(1-x^10)*(1-x^12)).

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5

Offset

0,25

Comments

Number of partitions of n into parts 7, 10, and 12. - Hoang Xuan Thanh, Sep 27 2025

Links

Table of n, a(n) for n=0..79.

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

Formula

a(n) = floor((n^2+36*n+288 - 7*(2*n-29)*(n mod 2))/1680 + ((3*n^2+3*n+3) mod 7)/7 + ((4*n^2+4*n+2 + 4*(n mod 2)*(n+2)) mod 5)/5). - Hoang Xuan Thanh, Sep 27 2025

Mathematica

CoefficientList[Series[1/((1-x^7)(1-x^10)(1-x^12)), {x, 0, 100}], x] (* or *) LinearRecurrence[ {0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1}, 100] (* Harvey P. Dale, Apr 22 2023 *)

Prog

(PARI) a(n) = ((n^2+36*n+288 - 7*(2*n-29)*(n%2))/1680 + ((3*n^2+3*n+3)%7)/7 + ((4*n^2+4*n+2 + 4*(n%2)*(n+2))%5)/5)\1 \\ Hoang Xuan Thanh, Sep 27 2025

Crossrefs

Sequence in context: A025914 A376631 A284977 * A212211 A321764 A333809

Adjacent sequences: A025913 A025914 A025915 * A025917 A025918 A025919

Keyword

nonn

Author

N. J. A. Sloane

Status

approved