A238867 Number of partitions of n where the difference between consecutive parts is at most 7.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 95, 125, 162, 210, 268, 344, 434, 549, 688, 861, 1069, 1328, 1637, 2016, 2472, 3023, 3682, 4479, 5424, 6558, 7905, 9508, 11404, 13657, 16307, 19440, 23123, 27454, 32526, 38479, 45424, 53545, 63006, 74024, 86824, 101701, 118931, 138899, 161983, 188656, 219419, 254895, 295709

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 seven 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^(8*i))/(1-q^i) ) ).

a(n) = Sum_{k=0..7} A238353(n,k). - Alois P. Heinz, Mar 09 2014

a(n) ~ 7^(1/4) * exp(Pi*sqrt(7*n/12)) / (2^(7/2) * 3^(1/4) * 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(7, 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[7, 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^(8*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), A238866 (d=6), this sequence, A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

Sequence in context: A008638 A008632 A347575 * A035988 A088669 A091580

Adjacent sequences: A238864 A238865 A238866 * A238868 A238869 A238870

Keyword

nonn

Author

Joerg Arndt, Mar 08 2014

Status

approved