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

0,3

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A000110, A102759.

Sequence in context: A003724 A138314 A115277 * A036782 A050237 A050258

Adjacent sequences: A130218 A130219 A130220 * A130222 A130223 A130224

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 05 2007, Aug 05 2007

Extensions

More terms from Emeric Deutsch, Aug 28 2007

Status

approved