A029294 - OEIS

A029294

Expansion of 1/((1-x^3)*(1-x^6)*(1-x^7)*(1-x^10)).

1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 2, 0, 3, 3, 1, 3, 4, 2, 4, 5, 4, 5, 6, 5, 7, 7, 7, 9, 9, 8, 12, 11, 10, 14, 14, 12, 17, 17, 15, 19, 21, 18, 23, 24, 22, 26, 28, 26, 31, 32, 31, 35, 37, 35, 41, 42, 41, 46, 48, 46, 53, 54, 53, 59, 61, 59, 67, 68, 67, 74, 77, 74, 83

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

a(n) is the number of partitions of n into parts 3, 6, 7, and 10. - Joerg Arndt, Apr 07 2026

LINKS

Table of n, a(n) for n=0..72.

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,1,0,-1,0,0,0,-2,0,0,0,-1,0,1,1,0,0,1,0,0,-1).

FORMULA

G.f.: 1/((1-x^3)*(1-x^6)*(1-x^7)*(1-x^10)).

a(n) = a(n-3) + a(n-6) + a(n-7) - a(n-9) - 2*a(n-13) - a(n-17) + a(n-19) + a(n-20) + a(n-23) - a(n-26). - Wesley Ivan Hurt, Jun 20 2025

a(n) = floor((n^3+39*n^2+630*n+7201)/7560 - (n mod 2)*n/120 - ((2*n^2+n) mod 3)*n/18 - (n mod 3)/2 + ((-1)^floor(n/3) - 1)*((n+2) mod 3)/12 + ((3*n^3+5*n^2+1) mod 7)/7). - Hoang Xuan Thanh, Apr 07 2026

MATHEMATICA

CoefficientList[Series[1/((1-x^3)(1-x^6)(1-x^7)(1-x^10)), {x, 0, 100}], x] (* Jinyuan Wang, Mar 11 2020 *)

LinearRecurrence[{0, 0, 1, 0, 0, 1, 1, 0, -1, 0, 0, 0, -2, 0, 0, 0, -1, 0, 1, 1, 0, 0, 1, 0, 0, -1}, {1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 2, 0, 3, 3, 1, 3, 4, 2, 4, 5, 4, 5, 6, 5, 7, 7}, 70] (* Harvey P. Dale, Oct 18 2020 *)

PROG

(PARI) a(n) = floor((n^3+39*n^2+630*n+7201)/7560 - (n%2)*n/120 - ((2*n^2+n)%3)*n/18 - [0, 3, 6, 2, 3, 7][n%6+1]/6 + ((3*n^3+5*n^2+1)%7)/7) \\ Hoang Xuan Thanh, Apr 07 2026

CROSSREFS

Sequence in context: A281423 A256626 A135211 * A065434 A045832 A287528

Adjacent sequences: A029291 A029292 A029293 * A029295 A029296 A029297

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved