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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.

%I A097763 #10 Jan 10 2016 09:58:32
%S A097763 0,0,0,3,10,25,56,224,1506,9951,57992,315425,1761552,11022180,
%T A097763 78474748,603715831,4771273414,38070877273,309146434240,2598546954268,
%U A097763 22887194502518,211388690471531,2031261113410564,20121026325645745
%N 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.
%C A097763 a(n) = A000296(n) - A097762(n).
%H A097763 Alois P. Heinz, <a href="/A097763/b097763.txt">Table of n, a(n) for n = 1..500</a>
%F A097763 Exponential generating function: cosh(exp(x)-x-1).
%e A097763 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.
%p A097763 seq(coeff(series(cosh(exp(x)-x-1),x=0,25),x^i)*i!, i=1..24);
%p A097763 # second Maple program:
%p A097763 with(combinat):
%p A097763 b:= proc(n, i, t) option remember; `if`(n=0, t,
%p A097763       `if`(i<2, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A097763        b(n-i*j, i-1, irem(t+j, 2)), j=0..n/i)))
%p A097763     end:
%p A097763 a:= n-> b(n$2, 1):
%p A097763 seq(a(n), n=1..30);  # _Alois P. Heinz_, Mar 08 2015
%t A097763 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_ *)
%Y A097763 Cf. A000296, A097762.
%K A097763 easy,nonn
%O A097763 1,4
%A A097763 Isabel C. Lugo (izzycat(AT)gmail.com), Aug 23 2004