|
|
A089259
|
|
Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).
|
|
63
|
|
|
1, 1, 2, 4, 7, 12, 22, 36, 61, 101, 166, 267, 433, 686, 1088, 1709, 2671, 4140, 6403, 9824, 15028, 22864, 34657, 52288, 78646, 117784, 175865, 261657, 388145, 573936, 846377, 1244475, 1825170, 2669776, 3895833, 5671127, 8236945, 11936594, 17261557, 24909756
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Number of complete set partitions of the integer partitions of n. This is the Euler transform of A000009. If we change the combstruct command from unlabeled to labeled, then we get A000258. - Thomas Wieder, Aug 01 2008
Number of set multipartitions (multisets of sets) of integer partitions of n. Also a(n) < A270995(n) for n>5. - Gus Wiseman, Apr 10 2016
|
|
LINKS
|
|
|
EXAMPLE
|
The a(6) = 22 set multipartitions of integer partitions of 6:
(6) (15) (123) (12)(12) (1)(1)(1)(12) (1)(1)(1)(1)(1)(1)
(24) (1)(14) (1)(1)(13) (1)(1)(1)(1)(2)
(1)(5) (1)(23) (1)(2)(12)
(2)(4) (2)(13) (1)(1)(1)(3)
(3)(3) (3)(12) (1)(1)(2)(2)
(1)(1)(4)
(1)(2)(3)
(2)(2)(2)
(End)
|
|
MAPLE
|
with(combstruct): A089259:= [H, {H=Set(T, card>=1), T=PowerSet (Sequence (Z, card>=1), card>=1)}, unlabeled]; 1, seq (count (A089259, size=j), j=1..16); # Thomas Wieder, Aug 01 2008
# second Maple program:
with(numtheory):
b:= proc(n, i)
if n<0 or n>i*(i+1)/2 then 0
elif n=0 then 1
elif i<1 then 0
else b(n, i):= b(n-i, i-1) +b(n, i-1)
fi
end:
a:= proc(n) option remember; `if` (n=0, 1,
add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
end:
|
|
MATHEMATICA
|
max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 24 2014 *)
b[n_, i_] := b[n, i] = Which[n<0 || n>i*(i+1)/2, 0, n == 0, 1, i<1, 0, True, b[n-i, i-1] + b[n, i-1]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d* b[d, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)
|
|
PROG
|
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={concat([1], EulerT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)))} \\ Andrew Howroyd, Oct 26 2018
|
|
CROSSREFS
|
Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|