A225171 - OEIS

%I #19 Jun 24 2018 06:34:52

%S 2,4,4,32,24,8,416,304,96,16,7552,5440,1760,320,32,176128,125824,

%T 41280,8000,960,64,5018624,3566080,1180928,237440,31360,2688,128,

%U 168968192,119614464,39875584,8212736,1146880,111104,7168,256

%N Triangle read by rows: T(n,k), 1 <= k <= n, is the number of non-degenerate fanout-free Boolean functions of n variables having AND rank k.

%C Also the Bell transform of A225170. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016

%H J. P. Hayes, <a href="http://dx.doi.org/10.1145/321978.321988">Enumeration of fanout-free Boolean functions</a>, J. ACM, 23 (1976), 700-709.

%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>

%F Hayes (1976, Theorem 3) gives a recurrence.

%e Triangle begins

%e 2,

%e 4,4,

%e 32,24,8,

%e 416,304,96,16,

%e 7552,5440,1760,320,32,

%e 176128,125824,41280,8000,960,64,

%e 5018624,3566080,1180928,237440,31360,2688,128,

%e 168968192,119614464,39875584,8212736,1146880,111104,7168,256,

%e ...

%p # Function BellMatrix defined in A264428.

%p BellMatrix(n -> `if`(n=0,2,add(combinat:-eulerian2(n, k)*2^(2*n-k), k=0..n)), 9); # _Peter Luschny_, Jan 29 2016

%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t rows = 12;

%t M = BellMatrix[If[# == 0, 2, Sum[(#+k)!*Sum[(-1)^j/(k-j)!*Sum[(-1)^i*2^(# - i + j)*StirlingS1[# - i + j, j - i]/((# - i + j)!*i!), {i, 0, j}], {j, 1, k}], {k, 1, #}]]&, rows];

%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)

%Y Columns give A225170 (or A005172), A005756, A224767, A224768.

%K nonn,tabl

%O 1,1

%A _N. J. A. Sloane_, Apr 30 2013