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A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

%I A089259 #47 Jun 28 2025 19:25:17
%S A089259 1,1,2,4,7,12,22,36,61,101,166,267,433,686,1088,1709,2671,4140,6403,
%T A089259 9824,15028,22864,34657,52288,78646,117784,175865,261657,388145,
%U A089259 573936,846377,1244475,1825170,2669776,3895833,5671127,8236945,11936594,17261557,24909756
%N A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).
%C A089259 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 A089259 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 A089259 Alois P. Heinz, <a href="/A089259/b089259.txt">Table of n, a(n) for n = 0..10000</a>
%e A089259 From _Gus Wiseman_, Oct 22 2018: (Start)
%e A089259 The a(6) = 22 set multipartitions of integer partitions of 6:
%e A089259   (6)  (15)    (123)      (12)(12)      (1)(1)(1)(12)    (1)(1)(1)(1)(1)(1)
%e A089259        (24)    (1)(14)    (1)(1)(13)    (1)(1)(1)(1)(2)
%e A089259        (1)(5)  (1)(23)    (1)(2)(12)
%e A089259        (2)(4)  (2)(13)    (1)(1)(1)(3)
%e A089259        (3)(3)  (3)(12)    (1)(1)(2)(2)
%e A089259                (1)(1)(4)
%e A089259                (1)(2)(3)
%e A089259                (2)(2)(2)
%e A089259 (End)
%p A089259 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 A089259 # second Maple program:
%p A089259 with(numtheory):
%p A089259 b:= proc(n, i)
%p A089259       if n<0 or n>i*(i+1)/2 then 0
%p A089259     elif n=0 then 1
%p A089259     elif i<1 then 0
%p A089259     else b(n,i):= b(n-i, i-1) +b(n, i-1)
%p A089259       fi
%p A089259     end:
%p A089259 a:= proc(n) option remember; `if` (n=0, 1,
%p A089259        add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
%p A089259     end:
%p A089259 seq(a(n), n=0..100);  # _Alois P. Heinz_, Nov 11 2011
%t A089259 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 A089259 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 A089259 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A089259 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 A089259 Row sums of A285229 and of A360763.
%Y A089259 Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.
%K A089259 nonn
%O A089259 0,3
%A A089259 _N. J. A. Sloane_, Dec 23 2003