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A285854 Number of permutations of [n] with three ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285854 #13 May 30 2018 06:06:50
%S A285854 1,18,105,1005,6762,61572,558548,5807700,62757288,777291768,
%T A285854 9831740256,139111566048,2048834965824,32758018496640,545051532176640,
%U A285854 9812211976039680,182219827628874240,3627461543458659840,74765368810365696000,1632210845693218560000
%N A285854 Number of permutations of [n] with three ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285854 Alois P. Heinz, <a href="/A285854/b285854.txt">Table of n, a(n) for n = 3..450</a>
%H A285854 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A285854 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285854       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285854       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 4)
%p A285854     end:
%p A285854 a:= n-> coeff(b(n$2, 0), x, 3):
%p A285854 seq(a(n), n=3..25);
%t A285854 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285854 b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]], {x, 0, 4}];
%t A285854 a[n_] := Coefficient[b[n, n, 0], x, 3];
%t A285854 Table[a[n], {n, 3, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A285854 Column k=3 of A285849.
%Y A285854 Cf. A285918.
%K A285854 nonn
%O A285854 3,2
%A A285854 _Alois P. Heinz_, Apr 27 2017