A218700 Number of partitions of n in which any two distinct parts differ by at least 5.

1, 1, 2, 2, 3, 2, 4, 3, 6, 7, 9, 10, 15, 15, 19, 23, 26, 28, 36, 37, 48, 52, 62, 67, 85, 93, 110, 122, 144, 157, 194, 205, 241, 265, 304, 338, 391, 422, 483, 533, 607, 661, 760, 822, 933, 1032, 1151, 1260, 1432, 1554, 1751, 1920, 2137, 2333, 2621, 2848, 3176

Offset

0,3

Comments

Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 5 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^(5*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(-5*x)) dx = -0.9908078441778424729564846063206238729218368028... - Vaclav Kotesovec, Jan 28 2022

Example

a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].
a(7) = 3: [1,1,1,1,1,1,1], [1,6], [7].
a(8) = 6: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,1,6], [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-5), 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, 5];
a /@ Range[0, 70] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Crossrefs

Column k=5 of A218698.

Cf. A160975.

Sequence in context: A307995 A061889 A240089 * A325331 A333108 A266935

Adjacent sequences: A218697 A218698 A218699 * A218701 A218702 A218703

Keyword

nonn

Author

Alois P. Heinz, Nov 04 2012

Status

approved