A218703 Number of partitions of n in which any two distinct parts differ by at least 8.

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 4, 10, 7, 12, 13, 17, 16, 23, 21, 30, 30, 34, 35, 47, 43, 51, 52, 66, 63, 81, 77, 100, 99, 120, 121, 156, 150, 185, 189, 234, 230, 283, 281, 343, 350, 409, 414, 503, 497, 587, 600, 695, 703, 824, 830, 967, 988, 1122, 1148, 1333

Offset

0,3

Comments

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 8 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^(8*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(-8*x)) dx = -1.1447921975208768146551512630331558734964408879... - Vaclav Kotesovec, Jan 28 2022

Example

a(9) = 3: [1,1,1,1,1,1,1,1,1], [3,3,3], [9].
a(10) = 5: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,9], [10].
a(11) = 4: [1,1,1,1,1,1,1,1,1,1,1], [1,1,9], [1,10], [11].

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-8), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

Crossrefs

Column k=8 of A218698.

Cf. A160978.

Sequence in context: A169819 A373738 A134681 * A326641 A144372 A182861

Adjacent sequences: A218700 A218701 A218702 * A218704 A218705 A218706

Keyword

nonn

Author

Alois P. Heinz, Nov 04 2012

Status

approved