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

0,3

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).

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

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 *)

Prog

(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

Crossrefs

Column k=5 of A194621.

Sequence in context: A319472 A357660 A309099 * A339672 A026813 A008636

Adjacent sequences: A218504 A218505 A218506 * A218508 A218509 A218510

Keyword

nonn,easy

Author

Alois P. Heinz, Oct 31 2012

Status

approved