OFFSET
0,5
COMMENTS
Number of partitions of n into parts 2, 3, 4, and 12. - Stefano Spezia, May 06 2023
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1,0,0,1,0,-1,-1,-1,1,1,1,0,-1).
FORMULA
a(n) = floor((n^3+36*n^2+364*n+1728)/1728 - (n^2+21*n+21)*(n mod 2)/192 + n*(((n+2) mod 3)-(n mod 3))/108 + n*((n^2+n+2) mod 4)/96). - Hoang Xuan Thanh, Sep 30 2025
MATHEMATICA
CoefficientList[Series[1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^12)), {x, 0, 70}], x] (* G. C. Greubel, Oct 29 2025 *)
LinearRecurrence[{0, 1, 1, 1, -1, -1, -1, 0, 1, 0, 0, 1, 0, -1, -1, -1, 1, 1, 1, 0, -1}, {1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 8, 5, 9, 8, 12, 9, 15, 12, 18}, 70] (* Harvey P. Dale, Dec 29 2025 *)
PROG
(PARI) Vec(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^12)) + O(x^99)) \\ Jinyuan Wang, Mar 18 2020
(PARI) a(n) = (n^3+36*n^2+364*n+1728 - 9*(n^2+21*n+21)*(n%2) + 16*n*(((n+2)%3)-(n%3)) + 18*n*((n^2+n+2)%4))\1728 \\ Hoang Xuan Thanh, Sep 30 2025
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 70);
Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^12)) )); // G. C. Greubel, Oct 29 2025
(SageMath)
def A029142_list(prec):
P.<x>= PowerSeriesRing(QQ, prec)
return P( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^12)) ).list()
print(A029142_list(71)) # G. C. Greubel, Oct 29 2025
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Stefano Spezia, May 06 2023
STATUS
approved