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

0,3

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..9} (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, 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 *)

Crossrefs

Column k=9 of A194621.

Sequence in context: A036010 A328545 A192061 * A008640 A008634 A347577

Adjacent sequences: A218508 A218509 A218510 * A218512 A218513 A218514

Keyword

nonn

Author

Alois P. Heinz, Oct 31 2012

Status

approved