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A368678 Number of permutations of [n] whose cycle maxima sum to 2n.

%I A368678 #19 Oct 03 2024 07:55:24
%S A368678 1,0,0,1,2,10,41,260,1552,12818,101280,1021908,10154064,121656672,
%T A368678 1447205472,20215013184,280271024640,4457067906240,70826580095040,
%U A368678 1264147627392000,22588177271650560,448332829478760960,8899910723677639680,194096853444946636800
%N A368678 Number of permutations of [n] whose cycle maxima sum to 2n.
%H A368678 Alois P. Heinz, <a href="/A368678/b368678.txt">Table of n, a(n) for n = 0..451</a>
%H A368678 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A368678 a(n) = A143947(n,2n).
%e A368678 a(0) = 1: the empty permutation.
%e A368678 a(3) = 1: (1)(2)(3).
%e A368678 a(4) = 2: (1)(23)(4), (1)(24)(3).
%e A368678 a(5) = 10: (12)(3)(45), (13)(2)(45), (1)(234)(5), (1)(243)(5), (1)(235)(4),
%e A368678   (1)(253)(4), (145)(2)(3), (154)(2)(3), (1)(24)(35), (1)(25)(34).
%p A368678 b:= proc(n) option remember;
%p A368678       `if`(n=0, 1, expand(b(n-1)*(t-n+x^n)))
%p A368678     end:
%p A368678 a:= n-> coeff(subs(t=n, b(n)), x, 2*n):
%p A368678 seq(a(n), n=0..23);
%t A368678 T[n_] := Module[{t}, CoefficientList[Product[n-k+t^k, {k, 1, n-1}]*t^(n-1), t]];
%t A368678 a[n_] := Switch[n, 0, 1, 1|2, 0, _, T[n][[2 n]]];
%t A368678 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Oct 03 2024 *)
%Y A368678 Cf. A143947, A367594, A368246, A368675.
%K A368678 nonn
%O A368678 0,5
%A A368678 _Alois P. Heinz_, Jan 02 2024