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A345189 Number of rows with the value "false" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.

%I A345189 #10 May 21 2022 08:28:57
%S A345189 1,1,6,41,330,2882,26604,255313,2521986,25473638,261898548,2731724778,
%T A345189 28836047844,307477681188,3306988334808,35833139582529,
%U A345189 390803960909106,4286644113507902,47258491871201508,523372307883323566,5819831138546794860,64954314678710555612,727371707764232349672
%N A345189 Number of rows with the value "false" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.
%H A345189 G. C. Greubel, <a href="/A345189/b345189.txt">Table of n, a(n) for n = 1..925</a>
%H A345189 Volkan Yildiz, <a href="https://arxiv.org/abs/2106.04728">Notes on algebraic structure of truth tables of bracketed formulae connected by implications</a>, arXiv:2106.04728 [math.CO], 2021.
%F A345189 G.f.: (-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6.
%F A345189 a(n) = 2*A005159(n-1) - A345190(n). - _G. C. Greubel_, May 20 2022
%t A345189 CoefficientList[Series[(-2 -Sqrt[1-12*x] +Sqrt[5 +24*x +4*Sqrt[1-12*x]])/6, {x, 0, 40}], x]//Rest (* _G. C. Greubel_, May 20 2022 *)
%o A345189 (PARI)  my(x='x+O('x^30)); Vec((-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6)
%o A345189 (SageMath)
%o A345189 def A345189_list(prec):
%o A345189     P.<x> = PowerSeriesRing(ZZ, prec)
%o A345189     return P( (-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6 ).list()
%o A345189 a=A345189_list(40); a[1:] # _G. C. Greubel_, May 20 2022
%Y A345189 Cf. A005159 (unknown rows, shifted), A025226 (all rows), A345190 (true rows).
%K A345189 nonn
%O A345189 1,3
%A A345189 _Michel Marcus_, Jun 10 2021