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

1,3

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

Cf. A000108, A186997, A192479.

Sequence in context: A059724 A000708 A027248 * A020090 A020036 A294312

Adjacent sequences: A192478 A192479 A192480 * A192482 A192483 A192484

Keyword

nonn

Author

Volkan Yildiz, Jul 01 2011

Extensions

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

Status

approved