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 A344532 Number of cycle-up-down permutations of [n^2] having n cycles. 3
 1, 1, 7, 14698, 51629528080, 914192102910317528125, 199979553262025879510473132453855232, 1131253316618666789979709230473744963049785439771172168, 309491168658231587025767619097898747214052900521443034546657433273562730332160 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS For the definition of cycle-up-down permutations see A186366. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..22 Wikipedia, Permutation FORMULA a(n) = (n^2)! * [x^(n^2) y^n] 1/(1-sin(x))^y. a(n) = A186366(n^2,n). EXAMPLE a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23). MAPLE b:= proc(u, o) option remember; `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u)) end: g:= proc(n) option remember; expand(`if`(n=0, 1, add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n))) end: a:= n-> coeff(g(n^2), x, n): seq(a(n), n=0..9); MATHEMATICA b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]]; g[n_] := g[n] = Expand[If[n == 0, 1, Sum[g[n-j]*Binomial[n-1, j-1]*x*b[j-1, 0], {j, 1, n}]]]; a[n_] := Coefficient[g[n^2], x, n]; a /@ Range[0, 9] (* Jean-François Alcover, Jun 10 2021, after Alois P. Heinz *) CROSSREFS Cf. A186366, A218141, A344445. Sequence in context: A242773 A333338 A131676 * A280813 A203685 A134645 Adjacent sequences: A344529 A344530 A344531 * A344533 A344534 A344535 KEYWORD nonn AUTHOR Alois P. Heinz, May 22 2021 STATUS approved

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Last modified March 1 17:48 EST 2024. Contains 370442 sequences. (Running on oeis4.)