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%I #63 Jul 05 2023 03:02:46
%S 0,0,1,2,9,46,262,1588,10053,65686,439658,2999116,20774154,145726348,
%T 1033125004,7390626280,53281906861,386732675046,2823690230850,
%U 20725376703324,152833785130398,1131770853856100,8412813651862868
%N Number of truth tables of bracketed formulas (case 3).
%C Equals the self-convolution of A186997 (up to offset). - _Paul D. Hanna_, Jul 03 2023
%H G. C. Greubel, <a href="/A218045/b218045.txt">Table of n, a(n) for n = 0..1000</a>
%H Volkan Yildiz, <a href="http://arxiv.org/abs/1205.5595">General combinatorical structure of truth tables of bracketed formulas connected by implication</a>, arXiv:1205.5595 [math.CO], 2012.
%F 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.
%F 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
%F 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
%F a(n) ~ (2*sqrt(3)-3) * 2^(3*n-3) / (3 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 19 2014
%F From _Paul D. Hanna_, Jul 03 2023: (Start)
%F G.f. A(x) = Series_Reversion( x*(1 + sqrt(1 - 4*x - 4*x^2)) / 2 )^2.
%F 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)
%e 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 + ...
%t 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 *)
%t 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_ *)
%o (Maxima)
%o 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 */
%o (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
%Y Cf. A186997, A218182, A288470.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Oct 23 2012