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A344532 Number of cycle-up-down permutations of [n^2] having n cycles.

%I A344532 #19 Jun 10 2021 07:42:40
%S A344532 1,1,7,14698,51629528080,914192102910317528125,
%T A344532 199979553262025879510473132453855232,
%U A344532 1131253316618666789979709230473744963049785439771172168,309491168658231587025767619097898747214052900521443034546657433273562730332160
%N A344532 Number of cycle-up-down permutations of [n^2] having n cycles.
%C A344532 For the definition of cycle-up-down permutations see A186366.
%H A344532 Alois P. Heinz, <a href="/A344532/b344532.txt">Table of n, a(n) for n = 0..22</a>
%H A344532 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A344532 a(n) = (n^2)! * [x^(n^2) y^n] 1/(1-sin(x))^y.
%F A344532 a(n) = A186366(n^2,n).
%e A344532 a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).
%p A344532 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A344532       add(b(o-1+j, u-j), j=1..u))
%p A344532     end:
%p A344532 g:= proc(n) option remember; expand(`if`(n=0, 1,
%p A344532       add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
%p A344532     end:
%p A344532 a:= n-> coeff(g(n^2), x, n):
%p A344532 seq(a(n), n=0..9);
%t A344532 b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
%t A344532 g[n_] := g[n] = Expand[If[n == 0, 1,
%t A344532      Sum[g[n-j]*Binomial[n-1, j-1]*x*b[j-1, 0], {j, 1, n}]]];
%t A344532 a[n_] := Coefficient[g[n^2], x, n];
%t A344532 a /@ Range[0, 9] (* _Jean-François Alcover_, Jun 10 2021, after _Alois P. Heinz_ *)
%Y A344532 Cf. A186366, A218141, A344445.
%K A344532 nonn
%O A344532 0,3
%A A344532 _Alois P. Heinz_, May 22 2021