A218512 Number of partitions of n in which any two parts differ by at most 10.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 171, 223, 282, 362, 453, 572, 709, 884, 1084, 1337, 1626, 1984, 2394, 2896, 3468, 4162, 4951, 5897, 6972, 8249, 9696, 11402, 13330, 15586, 18131, 21090, 24417, 28264, 32580, 37541, 43097, 49449, 56544

Offset: 0

Author

Alois P. Heinz, Oct 31 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Column k=10 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, 10), 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, 10], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

Formula

G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..10} (1-x^(i+j)).