A218702 Number of partitions of n in which any two distinct parts differ by at least 7.

1, 1, 2, 2, 3, 2, 4, 2, 4, 4, 6, 6, 11, 10, 13, 16, 19, 20, 25, 27, 33, 34, 39, 41, 51, 52, 61, 65, 80, 82, 99, 104, 126, 133, 156, 168, 199, 209, 243, 261, 302, 320, 372, 392, 447, 479, 537, 572, 650, 693, 770, 829, 920, 982, 1096, 1169, 1306, 1396, 1541

Offset

0,3

Comments

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 7 times.

Links

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

Formula

G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(7*i)/(1-x^i)).

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-7*x)) dx = -1.104868234083422137620242346741601264555358762... - Vaclav Kotesovec, Jan 28 2022

Example

a(8) = 4: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8].
a(9) = 4: [1,1,1,1,1,1,1,1,1], [3,3,3], [1,8], [9].
a(10) = 6: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,1,8], [1,9], [10].

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-7), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

Crossrefs

Column k=7 of A218698.

Cf. A160977.

Sequence in context: A001616 A257599 A366620 * A363525 A324181 A297169

Adjacent sequences: A218699 A218700 A218701 * A218703 A218704 A218705

Keyword

nonn

Author

Alois P. Heinz, Nov 04 2012

Status

approved