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A285855 Number of permutations of [n] with four ordered cycles such that equal-sized cycles are ordered with increasing least elements.

%I A285855 #13 May 30 2018 07:17:56
%S A285855 1,40,430,6300,62601,706608,8985560,107911760,1439518696,20364348576,
%T A285855 304923257184,4772610024000,80570363703696,1409795519233536,
%U A285855 26263500315144192,511153327733815296,10464902116976779776,223154458395064842240,5010190272214829475840
%N A285855 Number of permutations of [n] with four ordered cycles such that equal-sized cycles are ordered with increasing least elements.
%H A285855 Alois P. Heinz, <a href="/A285855/b285855.txt">Table of n, a(n) for n = 4..450</a>
%H A285855 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%p A285855 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285855       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285855       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 5)
%p A285855     end:
%p A285855 a:= n-> coeff(b(n$2, 0), x, 4):
%p A285855 seq(a(n), n=4..25);
%t A285855 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285855 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 A285855 a[n_] := Coefficient[b[n, n, 0], x, 4];
%t A285855 Table[a[n], {n, 4, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A285855 Column k=4 of A285849.
%Y A285855 Cf. A285919.
%K A285855 nonn
%O A285855 4,2
%A A285855 _Alois P. Heinz_, Apr 27 2017