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
Alois P. Heinz, Table of n, a(n) for n = 0..10000
EXAMPLE
From Gus Wiseman, Oct 22 2018: (Start)
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:
seq(a(n), n=0..100); # Alois P. Heinz, Nov 11, 2011
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 23 2003
STATUS
approved