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

Author

N. J. A. Sloane, Oct 23 2012

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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.

Crossrefs

Cf. A186997, A218182, A288470.

Programs

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 *)

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

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)