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

1, 1, 1, 5, 5, 27, 117, 331, 1213, 6579, 47193, 140527, 1213841, 4617927, 48210879, 243443739, 2392565149, 10377087115, 125434781845, 725455816883, 8086277450629, 59694530600595, 614469256831895, 4650128350629285, 52385811781286769, 467607504075929863

Offset: 0

Author

Vladeta Jovovic, Aug 04 2007, Aug 05 2007

Keywords

• easy
• nonn
• changed

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A055922, A102759, A111723, A111724.

Programs

Maple

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
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or irem(j, 2)=1, multinomial(n, n-i*j, i$j)
       /j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

Mathematica

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 *)

Formula

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

Extensions

More terms from Emeric Deutsch, Sep 01 2007