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A306535 Number of permutations p of [2n] having no index i with |p(i)-i| = n.

%I A306535 #23 May 10 2022 14:43:36
%S A306535 1,1,9,265,14833,1334961,176214841,32071101049,7697064251745,
%T A306535 2355301661033953,895014631192902121,413496759611120779881,
%U A306535 228250211305338670494289,148362637348470135821287825,112162153835443422680893595673,97581073836835777732377428235481
%N A306535 Number of permutations p of [2n] having no index i with |p(i)-i| = n.
%C A306535 Also 0th term of the 2n-th forward differences of n!.
%H A306535 Alois P. Heinz, <a href="/A306535/b306535.txt">Table of n, a(n) for n = 0..225</a>
%H A306535 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A306535 a(n) = A306512(2n,n).
%F A306535 a(n) = (2n)! - A306675(n).
%F A306535 a(n) = KummerU(-2*n, -2*n, -1). - _Peter Luschny_, May 10 2022
%p A306535 b:= proc(n, k) b(n, k):= `if`(k=0, n!, b(n+1, k-1) -b(n, k-1)) end:
%p A306535 a:= n-> b(0, 2*n):
%p A306535 seq(a(n), n=0..23);
%p A306535 seq(simplify(KummerU(-2*n, -2*n, -1)), n=0..15); # _Peter Luschny_, May 10 2022
%t A306535 b[n_, k_] := b[n, k] = If[k == 0, n!, b[n + 1, k - 1] - b[n, k - 1]];
%t A306535 a[n_] := b[0, 2n];
%t A306535 a /@ Range[0, 23] (* _Jean-François Alcover_, Apr 02 2021, after _Alois P. Heinz_ *)
%Y A306535 Cf. A306512, A306675.
%Y A306535 Cf. A000142, A047920, A068106.
%K A306535 nonn
%O A306535 0,3
%A A306535 _Alois P. Heinz_, Feb 22 2019