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A130220 Number of partitions of n-set in which number of blocks of size k is odd (or zero) for every k.

%I A130220 #14 Aug 22 2025 19:18:11
%S A130220 1,1,1,5,5,27,117,331,1213,6579,47193,140527,1213841,4617927,48210879,
%T A130220 243443739,2392565149,10377087115,125434781845,725455816883,
%U A130220 8086277450629,59694530600595,614469256831895,4650128350629285,52385811781286769,467607504075929863
%N A130220 Number of partitions of n-set in which number of blocks of size k is odd (or zero) for every k.
%H A130220 Alois P. Heinz, <a href="/A130220/b130220.txt">Table of n, a(n) for n = 0..500</a>
%F A130220 E.g.f.: Product_{k>0} (1+sinh(x^k/k!)).
%e A130220 a(4)=5 because we have abcd, a|bcd, acd|b, abd|c and abc|d.
%p A130220 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 A130220 # second Maple program:
%p A130220 with(combinat):
%p A130220 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A130220       add(`if`(j=0 or irem(j, 2)=1, multinomial(n, n-i*j, i$j)
%p A130220        /j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130220     end:
%p A130220 a:= n-> b(n$2):
%p A130220 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 08 2015
%t A130220 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 A130220 Cf. A055922, A102759, A111723, A111724.
%K A130220 easy,nonn,changed
%O A130220 0,4
%A A130220 _Vladeta Jovovic_, Aug 04 2007, Aug 05 2007
%E A130220 More terms from _Emeric Deutsch_, Sep 01 2007