|
| |
|
|
A050342
|
|
Expansion of Product_{m>=1} (1+x^m)^A000009(m).
|
|
35
|
|
|
|
1, 1, 1, 3, 4, 7, 12, 19, 30, 49, 77, 119, 186, 286, 438, 670, 1014, 1528, 2300, 3437, 5119, 7603, 11241, 16564, 24343, 35650, 52058, 75820, 110115, 159510, 230522, 332324, 477994, 686044, 982519, 1404243, 2003063, 2851720, 4052429, 5748440, 8140007, 11507125
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.
|
|
|
LINKS
|
Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
|
|
|
FORMULA
|
Weigh transform of A000009.
|
|
|
EXAMPLE
|
4=(4)=(3)+(1)=(3+1)=(2+1)+(1).
From Gus Wiseman, Oct 11 2018: (Start)
a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:
{{1}} {{2}} {{3}} {{4}} {{5}} {{6}}
{{1,2}} {{1,3}} {{1,4}} {{1,5}}
{{1},{2}} {{1},{3}} {{2,3}} {{2,4}}
{{1},{1,2}} {{1},{4}} {{1,2,3}}
{{2},{3}} {{1},{5}}
{{1},{1,3}} {{2},{4}}
{{2},{1,2}} {{1},{1,4}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{2},{3}}
{{1},{2},{1,2}}
(End)
|
|
|
MAPLE
|
g:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..50); # Alois P. Heinz, May 19 2013
|
|
|
MATHEMATICA
|
g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)
nn=10; Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k], {k, nn}], {x, 0, n}], {n, 0, nn}] (* Gus Wiseman, Oct 11 2018 *)
|
|
|
CROSSREFS
|
Cf. A050343-A050350, A089254.
Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.
Row sums of A330462 and of A360764.
Sequence in context: A158237 A117950 A025047 * A293642 A214286 A340359
Adjacent sequences: A050339 A050340 A050341 * A050343 A050344 A050345
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Christian G. Bower, Oct 15 1999
|
|
|
STATUS
|
approved
|
| |
|
|
