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

Author

Alois P. Heinz, Nov 04 2012

Keywords

• nonn

Comments

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 4 times.

Examples

			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].
		

Links

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

Crossrefs

Column k=4 of A218698.

Cf. A160974.

Programs

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 *)

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