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A363453 Total number of blocks containing only even elements in all partitions of [n].

%I A363453 #21 Dec 08 2023 07:11:44
%S A363453 0,0,1,2,12,35,206,780,4949,22686,156972,837333,6301550,38122554,
%T A363453 310279615,2090641920,18293310174,135445359397,1267153412532,
%U A363453 10202944645270,101557600812015,881921432827544,9299499328238110,86508104545175503,962663031508255416
%N A363453 Total number of blocks containing only even elements in all partitions of [n].
%H A363453 Alois P. Heinz, <a href="/A363453/b363453.txt">Table of n, a(n) for n = 0..250</a>
%H A363453 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363453 a(n) = Sum_{k=0..floor(n/2)} k * A124422(n,k).
%F A363453 a(n) = A363434(n) - A363452(n).
%F A363453 a(2n) = A363452(2n).
%F A363453 a(2n+1) = A363452(2n+1) - A094577(n).
%e A363453 a(3) = 2 = 0 + 0 + 1 + 0 + 1 : 123, 12|3, 13|2, 1|23, 1|2|3.
%p A363453 b:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
%p A363453       add(Stirling2(i, k)*binomial(g, i)*
%p A363453       add(Stirling2(u, j)*j^(g-i), j=0..u), i=k..g)
%p A363453     end:
%p A363453 a:= n-> add(b(n, k)*k, k=0..floor(n/2)):
%p A363453 seq(a(n), n=0..25);
%p A363453 # second Maple program:
%p A363453 b:= proc(n, x, y, m) option remember; `if`(n=0, x,
%p A363453       `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
%p A363453       `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
%p A363453     end:
%p A363453 a:= n-> b(n, 0$3):
%p A363453 seq(a(n), n=0..25);
%t A363453 b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, x,
%t A363453     If[x+m > 0, b[n-1, y, x, m]*(x+m), 0] + b[n-1, y, x+1, m] +
%t A363453     If[y > 0, b[n-1, y-1, x, m+1]*y, 0]];
%t A363453 a[n_] := b[n, 0, 0, 0];
%t A363453 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 08 2023, after _Alois P. Heinz_ *)
%Y A363453 Cf. A000110, A094577, A124422, A363434, A363452.
%K A363453 nonn
%O A363453 0,4
%A A363453 _Alois P. Heinz_, Jun 02 2023