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A285858 Number of permutations of [n] with seven ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285858 #13 May 30 2018 07:17:45
%S A285858 1,196,9114,330750,10094931,234138366,5932023097,142349568361,
%T A285858 3233779086538,74147737383720,1785843031638120,42966579274786440,
%U A285858 1047584220405271360,26222209747260881200,671966452779878874800,17944599541172975286000,485789620369911667323360
%N A285858 Number of permutations of [n] with seven ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285858 Alois P. Heinz, <a href="/A285858/b285858.txt">Table of n, a(n) for n = 7..450</a>
%H A285858 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A285858 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285858       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285858       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 8)
%p A285858     end:
%p A285858 a:= n-> coeff(b(n$2, 0), x, 7):
%p A285858 seq(a(n), n=7..25);
%t A285858 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285858 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, 8}];
%t A285858 a[n_] := Coefficient[b[n, n, 0], x, 7];
%t A285858 Table[a[n], {n, 7, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A285858 Column k=7 of A285849.
%Y A285858 Cf. A285922.
%K A285858 nonn
%O A285858 7,2
%A A285858 _Alois P. Heinz_, Apr 27 2017