A218699 Number of partitions of n in which any two distinct parts differ by at least 4.

1, 1, 2, 2, 3, 2, 5, 4, 8, 8, 12, 12, 19, 18, 24, 26, 36, 36, 48, 50, 70, 71, 92, 96, 129, 133, 168, 177, 225, 233, 294, 307, 382, 401, 488, 518, 635, 668, 803, 855, 1027, 1089, 1298, 1381, 1638, 1745, 2047, 2184, 2569, 2734, 3181, 3404, 3953, 4213, 4863, 5203

Offset

0,3

Comments

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 4 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^(4*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(-4*x)) dx = -0.9030055506558938921393786530232872470622617736... - Vaclav Kotesovec, Jan 28 2022

Example

a(5) = 2: [1,1,1,1,1], [5].
a(6) = 5: [1,1,1,1,1,1], [2,2,2], [3,3], [1,5], [6].
a(7) = 4: [1,1,1,1,1,1,1], [1,1,5], [1,6], [7].
a(8) = 8: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,1,1,5], [1,1,6], [2,6], [1,7], [8].

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-4), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i j, i - k, k], {j, 1, n/i}]]];
a[n_] := b[n, n, 4];
a /@ Range[0, 70] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Crossrefs

Column k=4 of A218698.

Cf. A160974.

Sequence in context: A330545 A113298 A058705 * A090794 A254858 A050323

Adjacent sequences: A218696 A218697 A218698 * A218700 A218701 A218702

Keyword

nonn

Author

Alois P. Heinz, Nov 04 2012

Status

approved