A218704 Number of partitions of n in which any two distinct parts differ by at least 9.

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 3, 8, 6, 9, 12, 14, 14, 20, 19, 24, 28, 29, 31, 41, 40, 44, 49, 54, 54, 69, 64, 77, 82, 94, 98, 119, 118, 139, 149, 173, 178, 215, 217, 253, 274, 306, 320, 375, 385, 440, 469, 521, 545, 626, 647, 718, 769, 838, 881, 994, 1026

Offset

0,3

Comments

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 9 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^(9*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(-9*x)) dx = -1.1777463281753839250026821894706693162991897931... - Vaclav Kotesovec, Jan 28 2022

Example

a(10) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
a(11) = 3: [1,1,1,1,1,1,1,1,1,1,1], [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-9), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

Crossrefs

Column k=9 of A218698.

Cf. A160979.

Sequence in context: A193459 A114102 A193513 * A144373 A086292 A335708

Adjacent sequences: A218701 A218702 A218703 * A218705 A218706 A218707

Keyword

nonn

Author

Alois P. Heinz, Nov 04 2012

Status

approved