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A032310 Number of ways to partition n labeled elements into sets of odd sizes, with all sizes different.

%I A032310 #19 Feb 05 2017 07:52:27
%S A032310 1,1,0,1,4,1,6,1,64,505,130,1321,1024,13157,2380,395851,5782144,
%T A032310 1639617,24545706,16100905,306621184,292018525,6304002100,1549052715,
%U A032310 507969498304,11794047630801,3164830777316,75389026652551,48756350408224,1240389053007865
%N A032310 Number of ways to partition n labeled elements into sets of odd sizes, with all sizes different.
%H A032310 Alois P. Heinz, <a href="/A032310/b032310.txt">Table of n, a(n) for n = 0..500</a>
%H A032310 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032310 "EGJ" (unordered, element, labeled) transform of 1, 0, 1, 0... (odds)
%F A032310 E.g.f.: Product_{k>0} (1+x^(2*k-1)/(2*k-1)!). - _Vladeta Jovovic_, Jan 16 2004
%p A032310 with(combinat):
%p A032310 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A032310       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
%p A032310     end:
%p A032310 a:= n-> b(n, iquo(n+1, 2)*2-1):
%p A032310 seq(a(n), n=0..40);  # _Alois P. Heinz_, Mar 08 2015
%t A032310 multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i-2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[n, Quotient[n+1, 2]*2-1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 05 2017, after _Alois P. Heinz_ *)
%Y A032310 Cf. A003724.
%K A032310 nonn
%O A032310 0,5
%A A032310 _Christian G. Bower_
%E A032310 Description corrected by _Vladeta Jovovic_, Aug 18 2004
%E A032310 a(0)=1 prepended by _Alois P. Heinz_, Mar 08 2015