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A285918 Number of ordered set partitions of [n] into three blocks such that equal-sized blocks are ordered with increasing least elements.

%I A285918 #12 May 17 2018 08:29:52
%S A285918 1,18,75,420,1218,4242,14563,42930,132528,432960,1250340,3814629,
%T A285918 12073701,35074482,106044555,331913202,967193328,2917846758,
%U A285918 9062084298,26507831559,79848170823,246771097680,723922691700,2178960263415,6709005218503,19728686792637
%N A285918 Number of ordered set partitions of [n] into three blocks such that equal-sized blocks are ordered with increasing least elements.
%H A285918 Alois P. Heinz, <a href="/A285918/b285918.txt">Table of n, a(n) for n = 3..1000</a>
%H A285918 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A285918 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285918       (p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
%p A285918       [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 4)
%p A285918     end:
%p A285918 a:= n-> coeff(b(n$2, 0), x, 3):
%p A285918 seq(a(n), n=3..30);
%t A285918 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285918 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, 4}];
%t A285918 a[n_] := Coefficient[b[n, n, 0], x, 3];
%t A285918 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%Y A285918 Column k=3 of A285824.
%Y A285918 Cf. A285854.
%K A285918 nonn
%O A285918 3,2
%A A285918 _Alois P. Heinz_, Apr 28 2017