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

Author

Vladeta Jovovic, Feb 25 2006; corrected Mar 05 2006

Keywords

• easy
• nonn

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<=mEmeric Deutsch, Apr 19 2006

Examples

			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).
		

Links

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

Crossrefs

Cf. A034296.

Programs

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
# second Maple program:
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 *)

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

Extensions

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