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

0,5

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).

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)).

Example

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].

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 *)

Prog

(PARI) concat(0, vector(40, n, k=0; forpart(p=2*n, k++, , [5, 5]); k))
(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)))

Crossrefs

Cf. Similar sequences: A000212 (3 parts), A001477 (2 parts), A014126 (4 parts), A256310 (6 parts).

Sequence in context: A258030 A164787 A131205 * A058682 A081995 A291141

Adjacent sequences: A256306 A256307 A256308 * A256310 A256311 A256312

Keyword

nonn,easy

Author

Colin Barker, Mar 22 2015

Status

approved