A218045 Number of truth tables of bracketed formulas (case 3).

0, 0, 1, 2, 9, 46, 262, 1588, 10053, 65686, 439658, 2999116, 20774154, 145726348, 1033125004, 7390626280, 53281906861, 386732675046, 2823690230850, 20725376703324, 152833785130398, 1131770853856100, 8412813651862868

Offset

0,4

Comments

Equals the self-convolution of A186997 (up to offset). - Paul D. Hanna, Jul 03 2023

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

From Paul D. Hanna, Jul 03 2023: (Start)

G.f. A(x) = Series_Reversion( x*(1 + sqrt(1 - 4*x - 4*x^2)) / 2 )^2.

G.f. A(x) = exp( Sum_{n>=1} A288470(n) * x^n/n ), where A288470(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n,2*k). (End)

Example

G.f. A(x) = x^2 + 2*x^3 + 9*x^4 + 46*x^5 + 262*x^6 + 1588*x^7 + 10053*x^8 + 65686*x^9 + 439658*x^10 + ...

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, A288470.

Sequence in context: A181997 A020053 A114194 * A161798 A365855 A373312

Adjacent sequences: A218042 A218043 A218044 * A218046 A218047 A218048

Keyword

nonn

Author

N. J. A. Sloane, Oct 23 2012

Status

approved