A225170 - OEIS

%I #27 Jun 24 2018 06:38:41

%S 2,4,32,416,7552,176128,5018624,168968192,6563282944,288909131776,

%T 14212910809088,772776684683264,46017323176296448,2978458881388183552,

%U 208198894960190160896,15631251601179130462208

%N Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 1.

%C Apart from initial term, same as A005172, which is the main entry for this sequence.

%H Vaclav Kotesovec, <a href="/A225170/b225170.txt">Table of n, a(n) for n = 1..241</a>

%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.

%F G.f.: 1/Q(0) + 1, where Q(k)= 1 - 2*x*(k+1) - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 18 2013

%F a(n) ~ (log(2)-1/2)^(1/2 - n) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Oct 19 2016

%t max = 16; s = -ProductLog[-Exp[x-1/2]/2] + O[x]^max; Join[{2}, Drop[CoefficientList[s, x]*Range[0, max-1]!, 2]] (* _Jean-François Alcover_, Oct 18 2016 *)

%t a[1] = 2; a[n_] := (Sum[(n + k - 1)!*Sum[(-1)^j/(k - j)!*Sum[(-1)^i*2^(n - i + j - 1)*StirlingS1[n - i + j - 1, j - i]/((n - i + j - 1)!*i!), {i, 0, j}], {j, 1, k}], {k, 1, n - 1}]);

%t Array[a, 20] (* _Jean-François Alcover_, Jun 24 2018, after _Vladimir Kruchinin_ *)

%Y Cf. A005172. A column of A225171.

%K nonn

%O 1,1

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