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A345190 Number of rows with the value "true" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.

%I A345190 #9 May 21 2022 08:29:25
%S A345190 1,5,30,229,1938,17530,165852,1621133,16242474,165923854,1721675460,
%T A345190 18095802306,192256162740,2061367432212,22276538889912,
%U A345190 242387718986301,2653259550491034,29198054511893638,322835545567447092,3584671507685675894,39955514234936341980,446897274497509974508
%N A345190 Number of rows with the value "true" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.
%H A345190 G. C. Greubel, <a href="/A345190/b345190.txt">Table of n, a(n) for n = 1..925</a>
%H A345190 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 A345190 G.f.: (4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6.
%F A345190 a(n) = 2*A005159(n-1) - A345189(n). - _G. C. Greubel_, May 20 2022
%t A345190 CoefficientList[Series[(4 -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 A345190 (PARI) my(x='x+O('x^30)); Vec((4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6)
%o A345190 (SageMath)
%o A345190 def A345190_list(prec):
%o A345190     P.<x> = PowerSeriesRing(ZZ, prec)
%o A345190     return P( (4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6 ).list()
%o A345190 a=A345190_list(40); a[1:] # _G. C. Greubel_, May 20 2022
%Y A345190 Cf. A005159 (unknown rows, shifted), A025226 (all rows), A345189 (false rows).
%K A345190 nonn
%O A345190 1,2
%A A345190 _Michel Marcus_, Jun 10 2021