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A240800 Total number of occurrences of the pattern 1<2<3 in all preferential arrangements (or ordered partitions) of n elements.

%I A240800 #15 Jun 08 2015 06:32:59
%S A240800 0,0,1,28,570,10700,200235,3857672,77620788,1641549000,36576771165,
%T A240800 859032716740,21251178078702,553095031003060,15122143306215855,
%U A240800 433634860865610320,13020228528050054760,408687299328542444880,13389274565474007735009,457150279686453405468780
%N A240800 Total number of occurrences of the pattern 1<2<3 in all preferential arrangements (or ordered partitions) of n elements.
%C A240800 There are A000670(n) preferential arrangements of n elements - see A000670, A240763.
%C A240800 The number that avoid the pattern 1<2<3 is given in A226316.
%H A240800 Alois P. Heinz, <a href="/A240800/b240800.txt">Table of n, a(n) for n = 1..120</a>
%F A240800 a(n) ~ n! * n^3 / (72 * (log(2))^(n+1)). - _Vaclav Kotesovec_, May 03 2015
%p A240800 b:= proc(n, t, h) option remember; `if`(n=0, [1, 0], add((p-> p+
%p A240800       [0, p[1]*j*h/6])(b(n-j, t+j, h+j*t))*binomial(n, j), j=1..n))
%p A240800     end:
%p A240800 a:= n-> b(n, 0$2)[2]:
%p A240800 seq(a(n), n=1..25);  # _Alois P. Heinz_, Dec 08 2014
%t A240800 b[n_, t_, h_] := b[n, t, h] = If[n == 0, {1, 0}, Sum[Function[{p}, p + {0, p[[1]]*j*h/6}][b[n - j, t + j, h + j*t]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0, 0][[2]]; Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Jun 08 2015, after _Alois P. Heinz_ *)
%Y A240800 Cf. A000670, A240763, A240796-A240800, A226316.
%K A240800 nonn
%O A240800 1,4
%A A240800 _N. J. A. Sloane_, Apr 13 2014
%E A240800 a(8)-a(20) from _Alois P. Heinz_, Dec 08 2014