A097763 Number of different partitions of the set {1, 2, ..., n} into an even number of blocks such that each block contains at least 2 elements.

0, 0, 0, 3, 10, 25, 56, 224, 1506, 9951, 57992, 315425, 1761552, 11022180, 78474748, 603715831, 4771273414, 38070877273, 309146434240, 2598546954268, 22887194502518, 211388690471531, 2031261113410564, 20121026325645745

Offset: 1

Author

Isabel C. Lugo (izzycat(AT)gmail.com), Aug 23 2004

Keywords

• easy
• nonn

Comments

a(n) = A000296(n) - A097762(n).

Examples

			a(6)=25 since we can partition a set of six elements into two non-singleton blocks, either of sizes four and two (15 ways) or three and three (10 ways); a(6)=15+10=25.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000296, A097762.

Programs

Maple

seq(coeff(series(cosh(exp(x)-x-1),x=0,25),x^i)*i!, i=1..24);
# second Maple program:
with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i<2, 0, add(multinomial(n, n-i*j, i$j)/j!*
       b(n-i*j, i-1, irem(t+j, 2)), j=0..n/i)))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 08 2015

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i < 2, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!*b[n - i*j, i - 1, Mod[t + j, 2]], {j, 0, n/i}]]]; a[n_] := b[n, n, 1];  Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 10 2016, after Alois P. Heinz *)

Formula

Exponential generating function: cosh(exp(x)-x-1).