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A285923 Number of ordered set partitions of [n] into eight blocks such that equal-sized blocks are ordered with increasing least elements.

%I A285923 #10 May 17 2018 08:05:18
%S A285923 1,288,18600,649440,18650346,378728064,6346968056,99768480240,
%T A285923 1370094506209,17452476893280,204026690329800,2291047776886752,
%U A285923 24663963563727574,256637317406331648,2540192740448641960,24558666993552144288,233835181800425532162
%N A285923 Number of ordered set partitions of [n] into eight blocks such that equal-sized blocks are ordered with increasing least elements.
%H A285923 Alois P. Heinz, <a href="/A285923/b285923.txt">Table of n, a(n) for n = 8..700</a>
%H A285923 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A285923 b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
%p A285923       (p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
%p A285923       [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)), x, 9)
%p A285923     end:
%p A285923 a:= n-> coeff(b(n$2, 0), x, 8):
%p A285923 seq(a(n), n=8..30);
%t A285923 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285923 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, 9}];
%t A285923 a[n_] := Coefficient[b[n, n, 0], x, 8];
%t A285923 Table[a[n], {n, 8, 30}] (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%Y A285923 Column k=8 of A285824.
%Y A285923 Cf. A285859.
%K A285923 nonn
%O A285923 8,2
%A A285923 _Alois P. Heinz_, Apr 28 2017