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A115029 Number of partitions of n such that all parts, with the possible exception of the smallest, appear only once. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

# 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-Fran├žois 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

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Last modified May 18 13:52 EDT 2021. Contains 343995 sequences. (Running on oeis4.)