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%I #44 Feb 20 2023 15:35:10
%S 1,1,2,4,7,12,22,36,61,101,166,267,433,686,1088,1709,2671,4140,6403,
%T 9824,15028,22864,34657,52288,78646,117784,175865,261657,388145,
%U 573936,846377,1244475,1825170,2669776,3895833,5671127,8236945,11936594,17261557,24909756
%N Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).
%C 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
%C 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
%H Alois P. Heinz, <a href="/A089259/b089259.txt">Table of n, a(n) for n = 0..10000</a>
%e From _Gus Wiseman_, Oct 22 2018: (Start)
%e The a(6) = 22 set multipartitions of integer partitions of 6:
%e (6) (15) (123) (12)(12) (1)(1)(1)(12) (1)(1)(1)(1)(1)(1)
%e (24) (1)(14) (1)(1)(13) (1)(1)(1)(1)(2)
%e (1)(5) (1)(23) (1)(2)(12)
%e (2)(4) (2)(13) (1)(1)(1)(3)
%e (3)(3) (3)(12) (1)(1)(2)(2)
%e (1)(1)(4)
%e (1)(2)(3)
%e (2)(2)(2)
%e (End)
%p 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
%p # second Maple program:
%p with(numtheory):
%p b:= proc(n, i)
%p if n<0 or n>i*(i+1)/2 then 0
%p elif n=0 then 1
%p elif i<1 then 0
%p else b(n,i):= b(n-i, i-1) +b(n, i-1)
%p fi
%p end:
%p a:= proc(n) option remember; `if` (n=0, 1,
%p add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Nov 11, 2011
%t max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* _Jean-François Alcover_, Mar 24 2014 *)
%t 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_ *)
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o 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
%Y Row sums of A285229 and of A360763.
%Y Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Dec 23 2003
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