A245069 - OEIS

%I #26 Nov 28 2018 08:00:46

%S 1,3,15,153,3567,196833,25604415,7727833473,5321887813887,

%T 8241841773665793,28359559029362676735,214672167825864945784833,

%U 3545389677599343390373285887,126865139766788531002690137546753,9777654677248888055040528691642187775

%N Number of Markov blanket structures with n labeled nodes.

%H Alois P. Heinz, <a href="/A245069/b245069.txt">Table of n, a(n) for n = 1..78</a>

%H Shyam Visweswaran and Gregory F. Cooper, <a href="http://arxiv.org/abs/1407.2483">Counting Markov blanket structures</a>, arXiv:1407.2483 [stat.ML], 2014

%F a(1) = 1; for n > 1, a(n) = Sum_{q=0..n-1} Sum_{r=0..n-q-1} ((n-1)!/(q!*r!*s!))*2^(q*s)*2^(r*s) * A003024(s); n-1=q+r+s.

%t (* b = A003024 *) b[0] = b[1] = 1; b[n_] := b[n] = Sum[-(-1)^k Binomial[n, k] 2^(k(n-k)) b[n-k], {k, 1, n}];

%t a[n_] := Sum[Sum[s = n-q-r-1; ((n-1)!/(q! r! s!)) 2^(q s) 2^(r s) b[s], {r, 0, n-q-1}], {q, 0, n 1}];

%t Array[a, 15] (* _Jean-François Alcover_, Nov 28 2018 *)

%K nonn

%O 1,2

%A _Shyam Visweswaran_, Jul 10 2014