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A324621 Number of permutations p of [1+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(1+n)*[i

%I A324621 #13 Oct 28 2021 09:27:19
%S A324621 0,1,1,7,31,185,1275,10095,90109,895169,9793829,116998199,1515196619,
%T A324621 21143666585,316260079951,5047672782687,85623656678457,
%U A324621 1538245254809537,29176112648650441,582614412521648359,12217688610474042487,268445509189890555577
%N A324621 Number of permutations p of [1+n] such that n is the maximum of the number of elements in any integer interval [p(i)..i+(1+n)*[i<p(i)]].
%H A324621 Alois P. Heinz, <a href="/A324621/b324621.txt">Table of n, a(n) for n = 0..449</a>
%F A324621 a(n) = A000166(n+1) - A000179(n+1) for n < 0, a(0) = 0.
%p A324621 a:= proc(n) option remember; `if`(n<5, [0, 1$2, 7, 31][n+1],
%p A324621       ((2*n^4-3*n^3-2*n^2+n+4)*a(n-1) -(n^5-4*n^4+7*n^2+6*n-14)*
%p A324621        a(n-2) -(n^5-2*n^4-4*n^3+2*n^2+13*n-12)*a(n-3)-(n-2)*
%p A324621        (n^3+2*n^2+n-2)*a(n-4))/(n^3-n^2-2))
%p A324621     end:
%p A324621 seq(a(n), n=0..23);
%t A324621 menage[n_] := If[n == 0, 1, 2n Sum[(-1)^k Binomial[2n-k, k] (n-k)!/(2n-k), {k, 0, n}]];
%t A324621 a[n_] := If[n == 0, 0, Subfactorial[n+1] - menage[n+1]];
%t A324621 a /@ Range[0, 21] (* _Jean-François Alcover_, Oct 28 2021 *)
%Y A324621 Row n=1 of A324563 and column of A324564 (as array).
%Y A324621 Cf. A000166, A000179.
%K A324621 nonn
%O A324621 0,4
%A A324621 _Alois P. Heinz_, Mar 09 2019