A238866 Number of partitions of n where the difference between consecutive parts is at most 6.

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 71, 91, 121, 155, 202, 255, 328, 410, 520, 647, 810, 1000, 1244, 1525, 1879, 2293, 2804, 3401, 4135, 4988, 6028, 7241, 8701, 10404, 12447, 14818, 17645, 20931, 24822, 29334, 34658, 40817, 48052, 56416, 66190, 77471, 90621, 105756, 123338, 143555, 166956, 193815, 224828, 260352

Offset

0,3

Comments

Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most 6 times (by taking conjugates).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Formula

G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(7*i))/(1-q^i) ) ).

a(n) = Sum_{k=0..6} A238353(n,k). - Alois P. Heinz, Mar 09 2014

a(n) ~ exp(Pi*sqrt(4*n/7)) / (2 * 7^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=0..min(6, n/i))))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> add(g(n, k), k=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 09 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[6, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

Prog

(PARI) N=66;  q = 'q + O('q^N);
Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1, k-1, (1-q^(7*i))/(1-q^i) ) ) )

Crossrefs

Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), this sequence, A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

Sequence in context: A008637 A008631 A347574 * A035978 A319475 A319454

Adjacent sequences: A238863 A238864 A238865 * A238867 A238868 A238869

Keyword

nonn

Author

Joerg Arndt, Mar 08 2014

Status

approved