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A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

%I A224271 #34 May 17 2023 17:09:34
%S A224271 1,1,3,8,28,107,459,2151,10931,59700,348146,2155925,14112377,97266301,
%T A224271 703484851,5323515156,42040470092,345670438963,2953171501547,
%U A224271 26166317121747,240047041176843,2276607815242880,22290187889601330,225018607554567149,2339331996135377345
%N A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.
%H A224271 Alois P. Heinz, <a href="/A224271/b224271.txt">Table of n, a(n) for n = 1..576</a>
%F A224271 E.g.f. A(x) satisfies: A'(x) = B'(x)*C(x) where B(x) is the e.g.f. for A003724 and C(x) is the e.g.f. for A005046.
%F A224271 a(n) = Sum_{k=0..floor((n-1)/2)} (k+1)*A124322(n-1,k). - _Alois P. Heinz_, Apr 02 2013
%F A224271 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * Bell(n-2*k-1). - _Ilya Gutkovskiy_, Apr 10 2022
%F A224271 From _Alois P. Heinz_, May 17 2023: (Start)
%F A224271 a(n) = Sum_{k=0..n-1} (-1)^k * A283424(n-1,k).
%F A224271 a(n) mod 2 = A131719(n+1). (End)
%e A224271 a(4) = 8 because we have: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3},{4}}.
%p A224271 with(combinat):
%p A224271 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A224271       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A224271       b(n-i*j, i-1)*`if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
%p A224271     end:
%p A224271 a:= n-> (p-> add(coeff(p, x, i)*(i+1), i=0..degree(p)))(b(n-1$2)):
%p A224271 seq(a(n), n=1..15);  # _Alois P. Heinz_, Mar 08 2015
%p A224271 # second Maple program:
%p A224271 b:= proc(n, t, m) option remember; `if`(n=0, t, (m-1)*
%p A224271       b(n-1, t, m)+b(n-1, 1-t, m)+b(n-1, t, m+1))
%p A224271     end:
%p A224271 a:= n-> b(n-1, 1$2):
%p A224271 seq(a(n), n=1..25);  # _Alois P. Heinz_, May 17 2023
%t A224271 nn=25;Drop[Range[0,nn]!CoefficientList[Series[Integrate[Exp[Cosh[x]-1]D[ Exp[Sinh[x]],x],x],{x,0,nn}],x],1]
%Y A224271 Cf. A000110, A003724, A005046, A102286, A124322, A131719, A283424.
%K A224271 nonn
%O A224271 1,3
%A A224271 _Geoffrey Critzer_, Apr 02 2013