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A111724 Number of partitions of an n-set with an even number of blocks of size 1.

%I A111724 #19 May 11 2020 17:23:48
%S A111724 0,2,1,11,21,117,428,2172,10727,59393,345335,2143825,14038324,
%T A111724 96834090,700715993,5305041715,41910528809,344714251149,2945819805408,
%U A111724 26107419715988,239556359980239,2272364911439153,22252173805170347,224666265799310801,2335958333831561032
%N A111724 Number of partitions of an n-set with an even number of blocks of size 1.
%H A111724 Alois P. Heinz, <a href="/A111724/b111724.txt">Table of n, a(n) for n = 1..576</a>
%F A111724 E.g.f.: cosh(x)*exp(exp(x)-1-x).
%F A111724 More generally, e.g.f. for number of partitions of an n-set with an even number of blocks of size k is cosh(x^k/k!)*exp(exp(x)-1-x^k/k!).
%p A111724 b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
%p A111724       `if`(j=1, 1-t, t))*binomial(n-1, j-1), j=1..n))
%p A111724     end:
%p A111724 a:= n-> b(n, 1):
%p A111724 seq(a(n), n=1..30);  # _Alois P. Heinz_, May 10 2016
%t A111724 Rest[ Range[0, 24]! CoefficientList[ Series[ Cosh[x]Exp[Exp[x] - 1 - x], {x, 0, 23}], x]] (* _Robert G. Wilson v_ *)
%o A111724 (Python)
%o A111724 from sympy.core.cache import cacheit
%o A111724 from sympy import binomial
%o A111724 @cacheit
%o A111724 def b(n, t): return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1) for j in range(1, n + 1))
%o A111724 def a(n): return b(n, 1)
%o A111724 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Aug 10 2017
%Y A111724 Cf. A097514, A113235, A063083, A062282, A111723, A111752, A111753.
%K A111724 easy,nonn
%O A111724 1,2
%A A111724 _Vladeta Jovovic_, Nov 17 2005
%E A111724 More terms from _Robert G. Wilson v_, Nov 22 2005