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A285856 Number of permutations of [n] with five ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285856 #13 May 30 2018 06:14:42
%S A285856 1,75,1400,28700,431445,6061545,101969450,1511771250,24207257360,
%T A285856 398261143280,7126152051200,125804137586400,2407653784466640,
%U A285856 47180921713154640,978889889736934560,20912829727776156000,470414513170923511680,10891943467262968682880
%N A285856 Number of permutations of [n] with five ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285856 Alois P. Heinz, <a href="/A285856/b285856.txt">Table of n, a(n) for n = 5..450</a>
%H A285856 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A285856 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285856       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285856       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 6)
%p A285856     end:
%p A285856 a:= n-> coeff(b(n$2, 0), x, 5):
%p A285856 seq(a(n), n=5..25);
%t A285856 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285856 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, 6}];
%t A285856 a[n_] := Coefficient[b[n, n, 0], x, 5];
%t A285856 Table[a[n], {n, 5, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A285856 Column k=5 of A285849.
%Y A285856 Cf. A285920.
%K A285856 nonn
%O A285856 5,2
%A A285856 _Alois P. Heinz_, Apr 27 2017