%I #12 Nov 11 2015 10:03:34

%S 1,1,1,5,5,27,117,331,1213,6579,47193,140527,1213841,4617927,48210879,

%T 243443739,2392565149,10377087115,125434781845,725455816883,

%U 8086277450629,59694530600595,614469256831895,4650128350629285

%N Number of partitions of n-set in which number of blocks of size k is odd (or zero) for every k.

%H Alois P. Heinz, <a href="/A130220/b130220.txt">Table of n, a(n) for n = 0..500</a>

%F E.g.f.: Product_{k>0} (1+sinh(x^k/k!)).

%e a(4)=5 because we have abcd, a|bcd, acd|b, abd|c and abc|d.

%p g:=product(1+sinh(x^k/factorial(k)),k=1..30): gser:=series(g,x=0,28): seq(factorial(n)*coeff(gser,x,n),n=0..24); # _Emeric Deutsch_, Sep 01 2007

%p # second Maple program:

%p with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(`if`(j=0 or irem(j, 2)=1, multinomial(n, n-i*j, i$j)

%p /j!*b(n-i*j, i-1), 0), j=0..n/i)))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 08 2015

%t multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[j == 0 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)

%Y Cf. A055922, A102759, A111723, A111724.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Aug 04 2007, Aug 05 2007

%E More terms from _Emeric Deutsch_, Sep 01 2007