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.

1, 1, 6, 41, 330, 2882, 26604, 255313, 2521986, 25473638, 261898548, 2731724778, 28836047844, 307477681188, 3306988334808, 35833139582529, 390803960909106, 4286644113507902, 47258491871201508, 523372307883323566, 5819831138546794860, 64954314678710555612, 727371707764232349672

Offset: 1

Author

Michel Marcus, Jun 10 2021

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..925
• Volkan Yildiz, Notes on algebraic structure of truth tables of bracketed formulae connected by implications, arXiv:2106.04728 [math.CO], 2021.

Crossrefs

Cf. A005159 (unknown rows, shifted), A025226 (all rows), A345190 (true rows).

Programs

Mathematica

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 *)

PARI

my(x='x+O('x^30)); Vec((-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6)

SageMath

def A345189_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6 ).list()
a=A345189_list(40); a[1:] # G. C. Greubel, May 20 2022

Formula

G.f.: (-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6.

a(n) = 2*A005159(n-1) - A345190(n). - G. C. Greubel, May 20 2022