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A285859 Number of permutations of [n] with eight ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285859 #13 May 30 2018 07:18:22
%S A285859 1,288,19560,921360,37423914,1124673264,34065856396,1010435626200,
%T A285859 27564092244689,746494701977024,20568917530438368,575594436161070144,
%U A285859 15985318079107792576,452561731064312392320,12942265817549110947520,383915932720263224659840
%N A285859 Number of permutations of [n] with eight ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285859 Alois P. Heinz, <a href="/A285859/b285859.txt">Table of n, a(n) for n = 8..450</a>
%H A285859 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A285859 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285859       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285859       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 9)
%p A285859     end:
%p A285859 a:= n-> coeff(b(n$2, 0), x, 8):
%p A285859 seq(a(n), n=8..25);
%t A285859 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285859 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, 9}];
%t A285859 a[n_] := Coefficient[b[n, n, 0], x, 8];
%t A285859 Table[a[n], {n, 8, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A285859 Column k=8 of A285849.
%Y A285859 Cf. A285923.
%K A285859 nonn
%O A285859 8,2
%A A285859 _Alois P. Heinz_, Apr 27 2017