A192481 a(n) = Sum_{i=1..n-1} (2^i*C(i)-a(i)) * (2^(n-i)*C(n-i)-a(n-i)), a(1)=1, where C(i)=A000108(i-1) are Catalan numbers.

1, 1, 6, 29, 162, 978, 6156, 40061, 267338, 1819238, 12576692, 88079378, 623581332, 4455663876, 32090099352, 232711721757, 1697799727066, 12452943237342, 91774314536100, 679234371006982, 5046438870909244, 37623611703611452, 281391143518722728

Offset: 1

Author

Volkan Yildiz, Jul 01 2011

Keywords

• nonn

Comments

a(n) is the number of rows with the value false in the truth tables of all bracketed m-implication, case (i), with n distinct variables.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Volkan Yildiz, Counting false entries in truth tables of bracketed formulas connected by m-implication, arXiv:1203.4645 [math.CO], 2012.
• Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv:1205.5595 [math.CO], 2012.

Crossrefs

Cf. A000108, A186997, A192479.

Programs

Maple

C := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A192481 := proc(n) option remember; if n<=1 then n; else add( (2^i*C(i-1)-procname(i))*(2^(n-i)*C(n-i-1)-procname(n-i)), i=1..n-1) ; end if; end proc:

Mathematica

CoefficientList[Series[(2 - Sqrt[1 - 8*x] - Sqrt[3 - 4*x - 2*Sqrt[1 - 8*x]])/2, {x,0,50}], x] (* G. C. Greubel, Feb 12 2017 *)

PARI

x='x+O('x^50); Vec((2-sqrt(1-8*x)-sqrt(3-4*x-2*sqrt(1-8*x)))/2) \\ G. C. Greubel, Feb 12 2017

Formula

G.f.: (2 - sqrt(1-8*x) - sqrt(3 - 4*x - 2*sqrt(1-8*x)))/2.

For large n, a(n) is asymptotically (1-2/sqrt 10) * 2^(3n-2)/ sqrt(pi*n^3).

D-finite with recurrence 10*n*(n-1)*(n-2)*a(n) -(n-1)*(n-2)*(149*n-396)*a(n-1) +2*(n-2)*(244*n^2-1618*n+2517)*a(n-2) +4

*(76*n^3-696*n^2+2165*n-2289)*a(n-3) +16*(2*n-9)*(56*n^2-336*n+451)*a(n-4) -256*(n-5)*(2*n-9)*(2*n-11)*a(n-5)=0. - R. J. Mathar, Jun 19 2021

Extensions

a(0) removed from definition by Georg Fischer, Jun 19 2021