A115029 Number of partitions of n such that all parts, with the possible exception of the smallest, appear only once.

1, 1, 2, 3, 5, 6, 10, 12, 17, 22, 29, 36, 48, 59, 73, 93, 114, 139, 171, 207, 250, 304, 361, 432, 517, 613, 722, 856, 1005, 1178, 1382, 1612, 1875, 2184, 2528, 2927, 3386, 3900, 4486, 5159, 5916, 6772, 7749, 8843, 10078, 11482, 13048, 14811, 16805, 19026

Offset

0,3

Comments

Also number of partitions of n such that if k is the largest part, then k and all integers from 1 to some integer m, 0<=m<k, occur any number of times (if m = 0, then partition consists only of k's). Example: a(5)=6 because we have [5], [4,1], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1] ([3,2] does not qualify). - Emeric Deutsch, Apr 19 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Anders Claesson, Svante Linusson, Henning Ulfarsson, and Emil Verkama, Inversion monotonicity in subclasses of the 1324-avoiders, arXiv:2604.01143 [math.CO], 2026. See pp. 19 (Table 3), 25 (Prop. 5.14), 39 (Prop. 7.4).

Formula

G.f.: 1+Sum_{k>=1} x^k/(1-x^k)*Product_{i>=k+1} (1+x^i).

G.f.: 1+Sum_{k>=1} (x^k/(1-x^k)) * Sum_{m=0..k-1} x^(m*(m+1)/2) / Product_{i=1..m} (1-x^i). - Emeric Deutsch, Apr 19 2006

a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - Vaclav Kotesovec, Jun 15 2025

Example

a(5) = 6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1] ([2,2,1] does not qualify).

Maple

g:=1+sum(x^k/(1-x^k)*product(1+x^i, i=k+1..90), k=1..90): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..44); # Emeric Deutsch, Apr 19 2006
# Alternative:
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, b(n, i-1)+
      `if`(irem(n, i)=0, 1, 0)+`if`(n>i, b(n-i, i-1), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2019

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[Mod[n, i] == 0, 1, 0] + If[n > i, b[n - i, i - 1], 0]];
a[n_] := b[n, n];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 21 2020, after Alois P. Heinz *)

Crossrefs

Cf. A034296.

Sequence in context: A130900 A007211 A027593 * A241443 A023025 A130898

Adjacent sequences: A115026 A115027 A115028 * A115030 A115031 A115032

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 25 2006; corrected Mar 05 2006

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 03 2019

Status

approved