A029149 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^12)).

1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 6, 5, 7, 8, 8, 10, 11, 11, 14, 14, 16, 17, 20, 20, 23, 25, 26, 29, 32, 32, 37, 38, 41, 44, 48, 49, 54, 57, 60, 64, 69, 70, 77, 80, 84, 89, 95, 97, 105, 109, 114, 120, 127, 130, 139, 144, 150, 157, 166, 169, 180, 186

Offset

0,6

Comments

a(n) is the number of partitions of n into parts 2, 3, 5, and 12. [Joerg Arndt, Jan 10 2018]

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

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

Formula

G.f.: 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^12)).

a(n) = a(n-2)+a(n-3)-a(n-7)-a(n-8)+a(n-10)+a(n-12)-a(n-14)-a(n-15)+a(n-19)+a(n-20)-a(n-22). - Wesley Ivan Hurt, May 20 2021

a(n) = floor((n^3+33*n^2+360*n+2160)/2160 - (n+5)*((n mod 2)/48 + ((2*n^2+2*n) mod 3)/36)). - Hoang Xuan Thanh, Oct 02 2025

Mathematica

CoefficientList[Series[1/((1-x^2)(1-x^3)(1-x^5)(1-x^12)), {x, 0, 60}], x] (* Harvey P. Dale, Jan 08 2018 *)
(* Alternative: *)
LinearRecurrence[{0, 1, 1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 0, -1, -1, 0, 0, 0, 1, 1, 0, -1}, {1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 6, 5, 7, 8, 8, 10, 11, 11, 14, 14}, 60] (* Harvey P. Dale, Jan 08 2018 *)

Prog

(Magma) m:=100; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^12)))); // Vincenzo Librandi, Jan 10 2018
(PARI) Vec(1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^12)) + O(x^70)) \\ Jinyuan Wang, Feb 28 2020
(PARI) a(n) = (n^3+33*n^2+360*n+2160 - 45*(n+5)*(n%2) - 60*(n+5)*((2*n^2+2*n)%3)) \2160 \\ Hoang Xuan Thanh, Oct 02 2025

Crossrefs

Sequence in context: A120179 A032739 A240868 * A080570 A163001 A239913

Adjacent sequences: A029146 A029147 A029148 * A029150 A029151 A029152

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Status

approved