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A285857 Number of permutations of [n] with six ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285857 #13 May 30 2018 06:14:33
%S A285857 1,126,3822,105336,2312163,41420610,868380535,16453085220,
%T A285857 312866654100,6063351173880,127050688947000,2603853165950400,
%U A285857 56141875342402480,1242418296237553440,28627706535786406800,683460419058369489600,16802904218347937067840
%N A285857 Number of permutations of [n] with six ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285857 Alois P. Heinz, <a href="/A285857/b285857.txt">Table of n, a(n) for n = 6..450</a>
%H A285857 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A285857 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285857       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285857       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 7)
%p A285857     end:
%p A285857 a:= n-> coeff(b(n$2, 0), x, 6):
%p A285857 seq(a(n), n=6..25);
%t A285857 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285857 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, 7}];
%t A285857 a[n_] := Coefficient[b[n, n, 0], x, 6];
%t A285857 Table[a[n], {n, 6, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A285857 Column k=6 of A285849.
%Y A285857 Cf. A285921.
%K A285857 nonn
%O A285857 6,2
%A A285857 _Alois P. Heinz_, Apr 27 2017