A218506 Number of partitions of n in which any two parts differ by at most 4.

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 34, 41, 54, 64, 81, 95, 118, 136, 165, 189, 226, 256, 301, 339, 395, 441, 507, 564, 644, 711, 804, 885, 995, 1089, 1215, 1326, 1473, 1600, 1766, 1914, 2105, 2272, 2486, 2678, 2921, 3136, 3406, 3650, 3954, 4225, 4560, 4865

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 (1,1,0,1,-3,-1,0,1,3,-1,0,-1,-1,1).

Crossrefs

Column k=4 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, 4), i=1..n):
seq(a(n), n=0..80);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k - 1] + b[n - i, i, k]]]];
a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 4], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

PARI

Vec((x^14-x^13-x^12+2*x^9-x^6-x^5+x^4-1)/((x-1)^5*(x+1)^3*(x^2+1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Mar 05 2015

Formula

G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..4} (1-x^(i+j)).

G.f.: (x^14-x^13-x^12+2*x^9-x^6-x^5+x^4-1) / ((x-1)^5*(x+1)^3*(x^2+1)^2*(x^2+x+1)). - Colin Barker, Mar 05 2015