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

Offset

0,4

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A055922, A102759, A111723, A111724.

Sequence in context: A220078 A302000 A219351 * A081847 A203124 A305180

Adjacent sequences: A130217 A130218 A130219 * A130221 A130222 A130223

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 04 2007, Aug 05 2007

Extensions

More terms from Emeric Deutsch, Sep 01 2007

Status

approved