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A285919 Number of ordered set partitions of [n] into four blocks such that equal-sized blocks are ordered with increasing least elements.

%I A285919 #14 May 17 2018 08:10:23
%S A285919 1,40,350,3080,17129,82488,464650,1901680,8357426,35701952,159721016,
%T A285919 627687060,2642405289,10712590392,45568675202,178738923440,
%U A285919 736145997686,2946913512648,12311241803256,48275516469180,197284995875314,786939537437440,3254422571085400
%N A285919 Number of ordered set partitions of [n] into four blocks such that equal-sized blocks are ordered with increasing least elements.
%H A285919 Alois P. Heinz, <a href="/A285919/b285919.txt">Table of n, a(n) for n = 4..700</a>
%H A285919 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A285919 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285919       (p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
%p A285919       [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 5)
%p A285919     end:
%p A285919 a:= n-> coeff(b(n$2, 0), x, 4):
%p A285919 seq(a(n), n=4..30);
%t A285919 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285919 b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[x^j*b[n - i*j, i - 1, p + j]*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2, {j, 0, n/i}]], {x, 0, 5}];
%t A285919 a[n_] := Coefficient[b[n, n, 0], x, 4];
%t A285919 Table[a[n], {n, 4, 30}] (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%Y A285919 Column k=4 of A285824.
%Y A285919 Cf. A285855.
%K A285919 nonn
%O A285919 4,2
%A A285919 _Alois P. Heinz_, Apr 28 2017