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A324361 Total number of occurrences of n in the (signed) displacement sets of all permutations of [2n] divided by n!.

%I A324361 #14 Oct 28 2021 09:27:13
%S A324361 0,1,5,49,679,12151,266321,6906257,206788751,7020426511,266464077769,
%T A324361 11180868467209,513915970996583,25678820830238759,1385874945753239969,
%U A324361 80341660921985676961,4979071555472111291551,328496221117149603559327,22987138271050177264124441
%N A324361 Total number of occurrences of n in the (signed) displacement sets of all permutations of [2n] divided by n!.
%H A324361 Alois P. Heinz, <a href="/A324361/b324361.txt">Table of n, a(n) for n = 0..366</a>
%H A324361 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A324361 a(n) = n! [x^n] (1-exp(-x))/(1-x)^(n+1).
%F A324361 a(n) = -1/n! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (2n-j)!.
%F A324361 a(n) = (8*n-12)*a(n-1) - (16*n^2-64*n+59)*a(n-2) - (4*n-10)*a(n-3) for n > 2.
%F A324361 a(n) = A324362(n,n) = A306234(2n,n).
%p A324361 a:= proc(s) option remember; `if`(n<3, (3*n-1)*n/2,
%p A324361       (8*n-12)*a(n-1)-(16*n^2-64*n+59)*a(n-2)-(4*n-10)*a(n-3))
%p A324361     end:
%p A324361 seq(a(n), n=0..20);
%t A324361 A[n_, k_] := -Sum[(-1)^j*Binomial[n, j]*(n+k-j)!, {j, 1, n}]/k!;
%t A324361 a[n_] := A[n, n];
%t A324361 a /@ Range[0, 20] (* _Jean-François Alcover_, Oct 28 2021, after _Alois P. Heinz_ in A324362 *)
%Y A324361 Main diagonal of A324362.
%Y A324361 Cf. A000142, A306234.
%K A324361 nonn
%O A324361 0,3
%A A324361 _Alois P. Heinz_, Feb 23 2019