A218507 Number of partitions of n in which any two parts differ by at most 5.

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 37, 48, 62, 78, 98, 121, 149, 181, 219, 262, 313, 370, 436, 510, 595, 690, 797, 916, 1050, 1198, 1364, 1545, 1747, 1968, 2212, 2479, 2771, 3089, 3437, 3814, 4226, 4669, 5151, 5670, 6232, 6837, 7487, 8185, 8936, 9739, 10602

Offset: 0

Author

Alois P. Heinz, Oct 31 2012

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,-2,-2,1,1,2,1,1,-2,-2,0,0,0,1,1,-1).

Crossrefs

Column k=5 of A194621.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 5), i=1..n):
seq(a(n), n=0..80);

Mathematica

LinearRecurrence[{1,1,0,0,0,-2,-2,1,1,2,1,1,-2,-2,0,0,0,1,1,-1},{1,1,2,3,5,7,11,15,21,28,37,48,62,78,98,121,149,181,219,262,313},60] (* Harvey P. Dale, Jan 18 2016 *)

PARI

Vec((x^20-x^19-x^18+x^15+x^14+x^13-x^12-x^11-x^10+x^7+x^6-x^5+1)/((x-1)^6*(x+1)^2*(x^2+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015

Formula

G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..5} (1-x^(i+j)).

G.f.: (x^20 -x^19 -x^18 +x^15 +x^14 +x^13 -x^12 -x^11 -x^10 +x^7 +x^6 -x^5 +1) / ((x -1)^6*(x +1)^2*(x^2 +1)*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)^2). - Colin Barker, Mar 05 2015