%I #16 Mar 16 2015 10:55:33
%S 1,1,4,13,62,311,1822,11593,80964,608833,4910786,42159239,383478988,
%T 3678859159,37087880754,391641822541,4319860660448,49647399946049,
%U 593217470459314,7354718987639959,94445777492433516,1254196823154143191,17198114810490326714
%N Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 2 for all x.
%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="/A135312/b135312.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{i=0..floor(n/2)} C(n,2*i) * A006882(2*i-1) * A000248(n-2*i).
%F a(n) = A135302(n,2).
%F E.g.f.: exp (x*exp(x) + x^2/2).
%e a(2) = 4 because there are 4 relations of the given kind for 2 elements: 1R1, 2R2; 1R1, 2R2, 1R2; 1R1, 2R2, 2R1; 1R1, 2R2, 1R2, 2R1.
%p df:= proc(n) option remember; `if`(n<=1, 1, n*df(n-2)) end: u:= proc(n) add(binomial(n, i) *(n-i)^i, i=0..n) end: a:= proc(n) add(binomial(n, i+i) *df(i+i-1) *u(n-i-i), i=0..floor(n/2)) end: seq(a(n), n=0..50);
%t a[n_] := SeriesCoefficient[Exp[x*Exp[x] + x^2/2], {x, 0, n}]*n!; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 04 2014 *)
%Y Cf. A135302, A006882, A000248, A007318.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Dec 05 2007
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