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A218045 Number of truth tables of bracketed formulas (case 3). 2
0, 0, 1, 2, 9, 46, 262, 1588, 10053, 65686, 439658, 2999116, 20774154, 145726348, 1033125004, 7390626280, 53281906861, 386732675046, 2823690230850, 20725376703324, 152833785130398, 1131770853856100, 8412813651862868 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv:1205.5595 [math.CO], 2012.

FORMULA

Yildiz gives a g.f.: (2+2*sqrt(1-8*x)-(1+sqrt(1-8*x))*sqrt(2+2*sqrt(1-8*x)+8*x))/8.

a(n+1) = (Sum_{k = 0..n} (Sum_{i=0..n-k} (binomial(k, 2*k+i+1-n)*binomial(k+i-1, i)))*binomial(2*n,k))/n. - Vladimir Kruchinin, Nov 19 2014

G.f. G(x) = A(x)/x satisfies G(x) = x*((G(x)*(G(x)+1))/(1-G(x))+1)^2. - Vladimir Kruchinin, Nov 19 2014

a(n) ~ (2*sqrt(3)-3) * 2^(3*n-3) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 19 2014

MATHEMATICA

CoefficientList[Series[(2+2*Sqrt[1-8*x]-(1+Sqrt[1-8*x])*Sqrt[2+2*Sqrt[1-8*x]+8*x])/8, {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2014 after Yildiz *)

Flatten[{0, 0, Table[Sum[(Sum[Binomial[k, 2*k+i+2-n]*Binomial[k+i-1, i], {i, 0, n-k-1}]*Binomial[2*n-2, k])/(n-1), {k, 0, n-1}], {n, 2, 20}]}] (* Vaclav Kotesovec, Nov 19 2014 after Vladimir Kruchinin *)

PROG

(Maxima)

a(n):=sum((sum(binomial(k, 2*k+i-n)*binomial(k+i-1, i), i, 0, n-k+1))*binomial(2*n+2, k), k, 0, n+1)/(n+1); /* Vladimir Kruchinin, Nov 19 2014  */

(PARI) x='x+O('x^50); concat([0, 0], Vec((2+2*sqrt(1-8*x)-(1+sqrt(1-8*x))*sqrt(2 + 2*sqrt(1-8*x)+8*x))/8)) \\ G. C. Greubel, Apr 01 2017

CROSSREFS

Cf. A186997, A218182.

Sequence in context: A181997 A020053 A114194 * A161798 A134091 A219614

Adjacent sequences:  A218042 A218043 A218044 * A218046 A218047 A218048

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 23 2012

STATUS

approved

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Last modified April 16 09:04 EDT 2021. Contains 343030 sequences. (Running on oeis4.)