OFFSET
1,3
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Volkan Yildiz, Jul 01 2011
EXTENSIONS
a(0) removed from definition by Georg Fischer, Jun 19 2021
STATUS
approved