A029310 - OEIS

%I #12 Jun 30 2026 17:08:10

%S 1,0,0,1,0,0,1,1,0,2,2,0,2,2,1,2,3,2,3,4,3,4,4,4,5,5,5,7,7,6,9,8,7,10,

%T 10,9,12,13,11,14,15,13,16,17,16,19,20,19,22,23,22,25,26,25,29,30,29,

%U 33,34,33,37,38,37,42,43,42

%N Expansion of 1/((1-x^3)*(1-x^7)*(1-x^9)*(1-x^10)).

%C Number of partitions of n into parts 3, 7, 9, and 10. - _Hoang Xuan Thanh_, Apr 09 2026

%H <a href="/index/Rec#order_29">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,0,0,0,1,0,1,0,0,-1,-1,0,0,-1,-1,0,0,1,0,1,0,0,0,1,0,0,-1).

%F a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=0, a(5)=0, a(6)=1, a(7)=1, a(8)=0, a(9)=2, a(10)=2, a(11)=0, a(12)=2, a(13)=2, a(14)=1, a(15)=2, a(16)=3, a(17)=2, a(18)=3, a(19)=4, a(20)=3, a(21)=4, a(22)=4, a(23)=4, a(24)=5, a(25)=5, a(26)=5, a(27)=7, a(28)=7, a(n)=a(n-3)+a(n-7)+a(n-9)- a(n-12)-a(n-13)- a(n-16)-a(n-17)+a(n-20)+a(n-22)+a(n-26)-a(n-29). - _Harvey P. Dale_, Sep 09 2015

%F a(n) = floor((2*n^3+87*n^2+1422*n+14240)/22680 - ((2*n^2+n) mod 3)*n/27  - (n mod 3)/3 + ((2*n^2+1) mod 3)/9 + ((2*n^3+3*n^2+n+2) mod 7)/7 + ([(n mod 9)=1] - [(n mod 9)=6])/3). - _Hoang Xuan Thanh_, Apr 09 2026

%t CoefficientList[Series[1/((1-x^3)(1-x^7)(1-x^9)(1-x^10)),{x,0,70}],x] (* _Harvey P. Dale_, Sep 09 2015 *)

%t (* Alternative: *)

%t LinearRecurrence[{0,0,1,0,0,0,1,0,1,0,0,-1,-1,0,0,-1,-1,0,0,1,0,1,0,0,0,1,0,0,-1},{1,0,0,1,0,0,1,1,0,2,2,0,2,2,1,2,3,2,3,4,3,4,4,4,5,5,5,7,7},70] (* _Harvey P. Dale_, Sep 09 2015 *)

%o (PARI) Vec(1/((1-x^3)*(1-x^7)*(1-x^9)*(1-x^10)) + O(x^80)) \\ _Hoang Xuan Thanh_, Apr 09 2026

%K nonn,easy,changed

%O 0,10

%A _N. J. A. Sloane_