A218099 - OEIS

%I #13 Aug 02 2021 14:34:39

%S 7087261,451238935,22913136730,1087116745385,51075201835515,

%T 2437976801668408,119752042470064290,6093096859120003590,

%U 322215964319093498225,17735784941946000072572,1016521929886047797022408,60650840653136697085038930,3764766650086543657134295955

%N Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 9.

%C R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

%D A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

%H Alois P. Heinz, <a href="/A218099/b218099.txt">Table of n, a(n) for n = 9..200</a>

%F E.g.f.: t_9(x)-t_8(x), with t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) if k>=0 and t_k(x) = 0 else.

%F a(n) = A210917(n) - A210916(n).

%p t:= proc(k) option remember; `if`(k<0, 0,

%p       unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))

%p     end:

%p egf:= t(9)(x)-t(8)(x):

%p a:= n-> n!* coeff(series(egf, x, n+1), x, n):

%p seq(a(n), n=9..22);

%t m = 9; t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]] ; egf = t[m][x]-t[m-1][x]; a[n_] := n!*Coefficient[Series[egf, {x, 0, n+1}], x, n]; Table[a[n], {n, m, 22}] (* _Jean-François Alcover_, Feb 14 2014, after Maple *)

%Y Column k=9 of A135313.

%K nonn

%O 9,1

%A _Alois P. Heinz_, Oct 20 2012