A130221 Number of partitions of n-set in which number of blocks of size 2k is odd (or zero) for every k.

1, 1, 2, 5, 12, 37, 158, 667, 2740, 13461, 74710, 412095, 2406880, 15450541, 103187698, 715323395, 5236160612, 40014337437, 318488475658, 2637143123027, 22603231117364, 201268520010153, 1855401760331982, 17624602999352535, 173071602624629536

Offset: 0

Author

Vladeta Jovovic, Aug 05 2007, Aug 05 2007

Keywords

• easy
• nonn

Examples

			a(4)=12 because from the 15 (=A000110(4)) partitions of the 4-set {a,b,c,d} only the partitions ab|cd, ac|bd and ad|bc do not qualify.
		

Links

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

Crossrefs

Cf. A000110, A102759.

Programs

Maple

g:=exp(sinh(x))*(product(1+sinh(x^(2*k)/factorial(2*k)), k=1..25)): gser:= series(g,x=0,30): seq(factorial(n)*coeff(gser,x,n),n=0..23); # Emeric Deutsch, Aug 28 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(i, 2)=1 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[i, 2] == 1 || 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, Dec 22 2016, after Alois P. Heinz *)

Formula

E.g.f.: exp(sinh(x))*Product_{k>0} (1+sinh(x^(2*k)/(2*k)!)).

Extensions

More terms from Emeric Deutsch, Aug 28 2007