A256313 Number of partitions of 3n into exactly 4 parts.
0, 0, 2, 6, 15, 27, 47, 72, 108, 150, 206, 270, 351, 441, 551, 672, 816, 972, 1154, 1350, 1575, 1815, 2087, 2376, 2700, 3042, 3422, 3822, 4263, 4725, 5231, 5760, 6336, 6936, 7586, 8262, 8991, 9747, 10559, 11400, 12300, 13230, 14222, 15246, 16335, 17457
Offset: 0
Examples
For n=3 the 6 partitions of 3*3 = 9 are [1,1,1,6], [1,1,2,5], [1,1,3,4], [1,2,2,4], [1,2,3,3] and [2,2,2,3].
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).
Programs
-
Mathematica
LinearRecurrence[{2,0,-2,2,-2,0,2,-1},{0,0,2,6,15,27,47,72},60] (* Harvey P. Dale, Jul 18 2021 *) -
PARI
concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [4,4]); k))
-
PARI
concat([0,0], Vec(x^2*(x^2+2)*(x^2+x+1)/((x-1)^4*(x+1)^2*(x^2+1)) + O(x^100)))
Formula
G.f.: x^2*(x^2+2)*(x^2+x+1) / ((x-1)^4*(x+1)^2*(x^2+1)).
a(n) = (6*n^3+6*n^2-3*n-5+(3*n+1)*(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)+(-1)^((2*n+1-(-1)^n)/4)))/32. - Luce ETIENNE, Feb 17 2017