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%I #18 Sep 14 2017 10:33:41
%S 1,1,4,29,210,2116,25522,362832,6000276,113593688,2434603356,
%T 58523364604,1565365441708,46273309903536,1502773485741816,
%U 53336787604185656,2059209704215556448,86117458019804680576,3886421648246467359364,188615552477984650605744
%N Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 3 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="/A135429/b135429.txt">Table of n, a(n) for n = 0..150</a>
%H Alois P. Heinz, <a href="/A135429/a135429.jpg">Illustration with formula for a(n)</a>
%F a(n) = see program.
%p A025035:= proc(n) option remember; (3*n)! /n! /(6^n) end:
%p z:= proc(n) option remember; add(binomial(n,k+k) *doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2)) end:
%p r:= proc(n) option remember; n! * add(add(add(add(Stirling2(e,d) *a^(d+i) *(a*(a+1)/2)^(n-i-i-e-d-a) /a! /(n-i-i-e-d-a)! /i! /e! /(2^i), a=0..(n-i-i-e-d)), d=0..min(e,n-i-i-e)), e=0..(n-i-i)), i=0..floor(n/2)) end:
%p a:= proc(n) option remember; n! *add(add(A025035(i) *z(j) *r(n-3*i-j) /(3*i)! /j! /(n-3*i-j)!, j=0..(n-3*i)), i=0..floor(n/3)) end:
%p seq(a(n), n=0..30);
%t Unprotect[Power]; 0^0 = 1; A025035[n_] := A025035[n] = (3n)!/n!/6^n; z[n_] := z[n] = Sum[Binomial[n, k+k]*(k+k-1)!!*k^(n-k-k), {k, 0, Floor[n/2]}]; r[n_] := r[n] = n!*Sum[Sum[Sum[Sum[StirlingS2[e, d]*a^(d+i)*(a*(a+1)/2)^(n-i-i-e-d-a)/a!/(n-i-i-e-d-a)!/i!/e!/2^i, {a, 0, n-i-i-e-d}], {d, 0, Min[e, n-i-i-e]}], {e, 0, n-i-i}], {i, 0, Floor[n/2]}]; a[n_] := a[n] = n!*Sum[Sum[A025035[i]*z[j]*r[n-3*i-j]/(3i)!/j!/(n-3*i-j)!, {j, 0, n-3*i}], {i, 0, Floor[n/3]}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 05 2014, after _Alois P. Heinz_ *)
%Y Cf. A135312, A025035, A006882, A008277, A000142.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Dec 12 2007
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