A218508 Number of partitions of n in which any two parts differ by at most 6.

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 51, 69, 86, 112, 139, 176, 214, 268, 321, 394, 470, 567, 668, 800, 933, 1103, 1281, 1498, 1725, 2006, 2293, 2643, 3010, 3443, 3897, 4438, 4995, 5652, 6341, 7135, 7967, 8932, 9930, 11079, 12283, 13645, 15071, 16692, 18372

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..6} (1-x^(i+j)).

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, 6), 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, 6], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

Crossrefs

Column k=6 of A194621.

Sequence in context: A309194 A330640 A319474 * A340719 A026814 A008637

Adjacent sequences: A218505 A218506 A218507 * A218509 A218510 A218511

Keyword

nonn

Author

Alois P. Heinz, Oct 31 2012

Status

approved