A218511 Number of partitions of n in which any two parts differ by at most 9.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 130, 168, 216, 274, 348, 435, 544, 674, 831, 1017, 1244, 1507, 1823, 2193, 2629, 3136, 3734, 4420, 5223, 6148, 7215, 8438, 9850, 11453, 13292, 15382, 17758, 20447, 23502, 26935, 30818, 35181, 40082, 45570

Offset: 0

Author

Alois P. Heinz, Oct 31 2012

Keywords

• nonn

Links

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

Crossrefs

Column k=9 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, 9), 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, 9], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

Formula

G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..9} (1-x^(i+j)).