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%I #35 Jan 12 2022 21:30:59
%S 1,0,2,1,12,14,100,180,990,2310,10920,30030,129612,396576,1620168,
%T 5318841,21029580,72364578,280735884,997356360,3828988020,13905563100,
%U 53108050320,195875639310,746569720572,2784329809344,10610782107800
%N Number of bracketings of 0#0#0#...#0 giving result 0, where 0#0 = 1, 0#1 = 1#0 = 1#1 = 0.
%C Operation # can be interpreted as NOT OR. The ratio a(n)/A000108(n-1) converges to sqrt(3)/3. Thanks to Soren Galatius Smith.
%C Essentially second column of A112519. - _Paul Barry_, Sep 09 2005
%H Michael De Vlieger, <a href="/A055392/b055392.txt">Table of n, a(n) for n = 1..1670</a>
%H Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F G.f.: (1/2)*(1 + sqrt(3 - 2*sqrt(1 - 4*x))).
%F The g.f. Z is also given by Z(x) = C(x)U(xC(x)), where U(x) = C(-x) and C is the g.f. of the Catalan numbers. - D. G. Rogers, Oct 20 2005
%F a(n) = Sum_{j=0..n} (1/n)*(-1)^(j-1)*C(2*n-j-1, n-j)*C(2*(j-1), j-1). - _Paul Barry_, Sep 09 2005, corrected by _Peter Bala_, Aug 19 2014
%F G.f. A(x) satisfies: A(x) = x + 2*A(x)^3 + A(x)^4; thus, A(x - 2*x^3 - x^4) = x. - _Paul D. Hanna_, Apr 05 2012
%F G.f. A(x) satisfies: x = Sum_{n>=1} 1/(1+A(x))^(2*n-1) * Product_{k=1..n} (1 - 1/(1+A(x))^k). - _Paul D. Hanna_, Apr 05 2012
%F Conjecture: 500*n*(n-1)*a(n) +100*(n-1)*(5*n -12)*a(n-1) +20*(25*n^2 -463*n +846)*a(n-2) +(-140161*n^2 +966559*n -1637508)*a(n-3) +2*(250*n^2 -26509*n +105084)*a(n-4) +98036*(4*n -19)*(4*n -21)*a(n-5) = 0. - _R. J. Mathar_, Nov 26 2012
%F a(n) ~ 4^(n-1) / (sqrt(3*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 03 2019
%t CoefficientList[ Series[1/2 + 1/2(3 - 2(1 - 4x)^(1/2))^(1/2), {x, 0, 27}], x] (* _Robert G. Wilson v_, May 04 2004 *)
%o (PARI) {a(n)=if(n<1,0,polcoeff(serreverse(x - 2*x^3 - x^4 +x*O(x^n)),n))} /* _Paul D. Hanna_, Apr 05 2012 */
%o (Sage) [(1/n)*sum( (-1)^j*binomial(2*j,j)*binomial(2*n-j-2,n-j-1) for j in (0..n-1) ) for n in (1..30)] # _G. C. Greubel_, Jan 12 2022
%o (Magma) [(1/n)*(&+[(-1)^j*Binomial(2*j,j)*Binomial(2*n-j-2,n-j-1): j in [0..n-1]]): n in [1..30]]; // _G. C. Greubel_, Jan 12 2022
%Y Cf. A000108, A055113, A055395, A112519, A112521.
%K nonn
%O 1,3
%A _Jeppe Stig Nielsen_, Jun 24 2000
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