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A115029
Number of partitions of n such that all parts, with the possible exception of the smallest, appear only once.
14
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
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-François Alcover, Nov 21 2020, after Alois P. Heinz *)
CROSSREFS
Cf. A034296.
Sequence in context: A130900 A007211 A027593 * A241443 A023025 A130898
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