%I #16 Feb 28 2017 22:55:10
%S 0,0,1,12,130,1500,18935,262248,3972612,65500200,1169398065,
%T 22494463860,464072915878,10225330604580,239720548513355,
%U 5959152063448080,156592569864940040,4337574220496785680,126329273251232688069,3859509516112803668220,123426111134706786806890
%N Total number of occurrences of the pattern 1=2=3 in all preferential arrangements (or ordered partitions) of n elements.
%C There are A000670(n) preferential arrangements of n elements - see A000670, A240763.
%C The number that avoid the pattern 1=2=3 is given in A080599.
%H Alois P. Heinz, <a href="/A240798/b240798.txt">Table of n, a(n) for n = 1..420</a>
%F a(n) ~ n! * n / (12 * (log(2))^(n-1)). - _Vaclav Kotesovec_, May 03 2015
%p b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
%p [0, p[1]*binomial(j, 3)])(b(n-j))*binomial(n, j), j=1..n))
%p end:
%p a:= n-> b(n)[2]:
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Dec 08 2014
%t b[n_] := b[n] = If[n==0, {1, 0}, Sum[Function[p, p+{0, p[[1]]*Binomial[j, 3]} ][b[n-j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n][[2]]; Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Feb 28 2017, after _Alois P. Heinz_ *)
%Y Cf. A000670, A240763, A240796-A240800, A080599.
%K nonn
%O 1,4
%A _N. J. A. Sloane_, Apr 13 2014
%E a(8)-a(21) from _Alois P. Heinz_, Dec 08 2014