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A096648 Number of partitions of an n-set with odd number of even blocks.

%I A096648 #20 Sep 03 2016 13:29:10
%S A096648 0,1,3,7,25,106,434,2045,10707,57781,338195,2115664,13796952,95394573,
%T A096648 692462671,5235101739,41436754261,341177640610,2915100624274,
%U A096648 25866987547865,237448494222575,2252995117706961,22078799199129799,222971522853648704,2319210969809731600
%N A096648 Number of partitions of an n-set with odd number of even blocks.
%H A096648 Alois P. Heinz, <a href="/A096648/b096648.txt">Table of n, a(n) for n = 1..500</a>
%F A096648 E.g.f.: exp(sinh(x))*sinh(cosh(x)-1).
%F A096648 a(2*n) = A024429(2*n) and a(2*n+1) = A024430(2*n+1). - _Jonathan Vos Post_, Oct 19 2005
%F A096648 a(n) = sum{k=0..n, if(mod(n-k,2)=1, A048993(n,k), 0)}. - _Paul Barry_, May 19 2006
%p A096648 with(combinat):
%p A096648 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1,
%p A096648       0, add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
%p A096648       irem(t+`if`(irem(i, 2)=0, j, 0), 2)), j=0..n/i)))
%p A096648     end:
%p A096648 a:= n-> b(n$2, 0):
%p A096648 seq(a(n), n=1..30);  # _Alois P. Heinz_, Mar 08 2015
%t A096648 multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, Mod[t+If[Mod[i, 2] == 0, j, 0], 2]], {j, 0, n/i}]]]; a[n_] := b[n, n, 2]; Table[ a[n], {n, 1, 30}] (* _Jean-François Alcover_, May 13 2015, after _Alois P. Heinz_ *)
%t A096648 With[{nn=30},Rest[CoefficientList[Series[Exp[Sinh[x]]Sinh[Cosh[x]-1], {x,0,nn}],x] Range[0,nn]!]] (* _Harvey P. Dale_, Sep 03 2016 *)
%Y A096648 Cf. A000701.
%Y A096648 Cf. A024429, A024430, A096647.
%K A096648 easy,nonn
%O A096648 1,3
%A A096648 _Vladeta Jovovic_, Aug 14 2004
%E A096648 More terms from _Emeric Deutsch_, Nov 16 2004