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A111753 Number of partitions of {1,..,n} into lists with an odd number of lists of size 1, where a list means an ordered subset, cf. A000262.

%I A111753 #37 Dec 01 2021 09:26:32
%S A111753 0,1,0,7,24,201,1560,14743,154896,1813969,23346000,327496071,
%T A111753 4970498280,81121077337,1416223931304,26328776843671,519178407998880,
%U A111753 10821355158998433,237677397895531296,5485802780426178439,132728552830731814200,3358841601972480225001
%N A111753 Number of partitions of {1,..,n} into lists with an odd number of lists of size 1, where a list means an ordered subset, cf. A000262.
%C A111753 a(n) + A111752(n) = A000262(n). - _David Wasserman_, Feb 11 2009
%H A111753 Alois P. Heinz, <a href="/A111753/b111753.txt">Table of n, a(n) for n = 0..444</a>
%F A111753 E.g.f.: sinh(x)*exp(x^2/(1-x)). More generally, e.g.f. for number of partitions of {1, 2, ...n} into lists with an odd number of lists of size k is sinh(x^k)*exp(x/(1-x)-x^k).
%F A111753 E.g.f.: sinh(x)*exp(x^2/(1-x))=1/2*Q(0);   Q(k)=1-((2x-1)^k)/( 1-x/(x-((2x-1)^k)*(k+1)*(1-x)/Q(k+1)));  (continued fraction). - _Sergei N. Gladkovskii_, Nov 17 2011
%F A111753 a(n) ~ (exp(1)-exp(-1)) * 2^(-3/2) * exp(2*sqrt(n)-n-3/2) * n^(n-1/4) * (1 + (43/48 - coth(1))/sqrt(n)). - _Vaclav Kotesovec_, Dec 01 2021
%p A111753 b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
%p A111753      `if`(j=1, 1-t, t))*binomial(n-1, j-1)*j!, j=1..n))
%p A111753     end:
%p A111753 a:= n-> b(n, 0):
%p A111753 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 10 2016
%t A111753 b[n_, t_] := b[n, t] = If[n==0, t, Sum[b[n-j, If[j==1, 1-t, t]]*Binomial[ n-1, j-1]*j!, {j, 1, n}]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 03 2017, after _Alois P. Heinz_ *)
%o A111753 (Python)
%o A111753 from sympy.core.cache import cacheit
%o A111753 from sympy import binomial, factorial as f
%o A111753 @cacheit
%o A111753 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)*f(j) for j in range(1, n + 1)])
%o A111753 def a(n): return b(n, 0)
%o A111753 print([a(n) for n in range(51)]) # _Indranil Ghosh_, Aug 10 2017
%Y A111753 Cf. A113235, A063083, A062282, A111723, A111724, A111752.
%K A111753 easy,nonn
%O A111753 0,4
%A A111753 _Vladeta Jovovic_, Nov 19 2005; corrected Jun 06 2006
%E A111753 More terms from _David Wasserman_, Feb 11 2009
%E A111753 a(0)=0 prepended by _Alois P. Heinz_, May 10 2016