A218509 Number of partitions of n in which any two parts differ by at most 7.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 72, 93, 120, 153, 194, 242, 302, 372, 457, 556, 675, 812, 975, 1162, 1381, 1632, 1923, 2254, 2636, 3068, 3562, 4119, 4752, 5462, 6265, 7162, 8170, 9293, 10549, 11942, 13495, 15211, 17115, 19214, 21534, 24083, 26892

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..7} (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, 7), 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, 7], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

Crossrefs

Column k=7 of A194621.

Sequence in context: A209039 A182805 A309058 * A026815 A341913 A008638

Adjacent sequences: A218506 A218507 A218508 * A218510 A218511 A218512

Keyword

nonn

Author

Alois P. Heinz, Oct 31 2012

Status

approved