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A225170 Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 1.

%I A225170 #31 Mar 28 2025 22:53:54
%S A225170 2,4,32,416,7552,176128,5018624,168968192,6563282944,288909131776,
%T A225170 14212910809088,772776684683264,46017323176296448,2978458881388183552,
%U A225170 208198894960190160896,15631251601179130462208,1254492810303112820555776,107174403941451434687463424
%N A225170 Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 1.
%C A225170 Apart from initial term, same as A005172, which is the main entry for this sequence.
%H A225170 Vaclav Kotesovec, <a href="/A225170/b225170.txt">Table of n, a(n) for n = 1..241</a>
%H A225170 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 A225170 <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%F A225170 Hayes (1976, Theorem 3) gives a recurrence.
%F A225170 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 A225170 a(n) ~ (log(2)-1/2)^(1/2 - n) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Oct 19 2016
%F A225170 a(n) = 2^n * A000311(n). - _Andrew Howroyd_, Mar 28 2025
%t A225170 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 A225170 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 A225170 Array[a, 20] (* _Jean-François Alcover_, Jun 24 2018, after _Vladimir Kruchinin_ *)
%o A225170 (PARI) seq(n) = Vec(serlaplace(serreverse((1 + 2*x - exp(x + O(x*x^n)))/2 ))) \\ _Andrew Howroyd_, Mar 28 2025
%Y A225170 Column 1 of A225171.
%Y A225170 Cf. A000311, A005172.
%K A225170 nonn
%O A225170 1,1
%A A225170 _N. J. A. Sloane_, Apr 30 2013