A256309 Number of partitions of 2n into exactly 5 parts.
0, 0, 0, 1, 3, 7, 13, 23, 37, 57, 84, 119, 164, 221, 291, 377, 480, 603, 748, 918, 1115, 1342, 1602, 1898, 2233, 2611, 3034, 3507, 4033, 4616, 5260, 5969, 6747, 7599, 8529, 9542, 10642, 11835, 13125, 14518, 16019, 17633, 19366, 21224, 23212, 25337, 27604
Offset: 0
Examples
For n=4 the 3 partitions of 2*4 = 8 are [1,1,1,1,4], [1,1,1,2,3] and [1,1,2,2,2].
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,1,0,-1,1,1,0,-2,1).
Crossrefs
Programs
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Mathematica
CoefficientList[Series[- x^3 (x^4 + x^2 + x + 1) / ((x - 1)^5 (x + 1) (x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 22 2015 *) LinearRecurrence[{2,0,-1,-1,1,0,-1,1,1,0,-2,1},{0,0,0,1,3,7,13,23,37,57,84,119},50] (* Harvey P. Dale, Mar 06 2023 *) -
PARI
concat(0, vector(40, n, k=0; forpart(p=2*n, k++, , [5,5]); k))
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PARI
concat([0,0,0], Vec(-x^3*(x^4+x^2+x+1)/((x-1)^5*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))
Formula
G.f.: -x^3*(x^4+x^2+x+1) / ((x-1)^5*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)).