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A023053
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Number of noncrossing rooted trees with n nodes on a circle that do not have leaves at level 1.
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3
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1, 0, 2, 7, 34, 171, 905, 4952, 27802, 159254, 927081, 5468960, 32621669, 196422509, 1192294778, 7288208927, 44825586130, 277196752569, 1722454028174, 10749430579118, 67346519863117, 423425225290485, 2670741276559282, 16895070479910967, 107165705513319749, 681438064187707596
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{i=0,..,n} (-1)^i*(i+1)*binomial(3*n-2*i, n-i)/(2*n-i+1).
G.f.: g/(1+zg) where g = 1 + z*g^3, g(0) = 1.
G.f.: g/(1+zg) where g = 2*sin(arcsin(sqrt(27*z)/2)/3)/sqrt(3*z).
G.f.: A(x)=1/(1+x)*(1+6*x/((1+x)*G(0)-6*x)) ; G(k)= 3*x*(3*k+1)*(3*k+2) + (2*k+2)*(2*k+3) - 6*x*(k+1)*(2*k+3)*(3*k+4)*(3*k+5)/G(k+1) ; (continued fraction Euler's kind,1-step ). - Sergei N. Gladkovskii, Dec 29 2011
a(n) ~ 27^(n+3/2) / (121 * sqrt(Pi) * 4^(n+1) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence 2*n*(2*n+1)*a(n) +(-47*n^2+65*n-24)*a(n-1) +3*(49*n^2-167*n+148)*a(n-2) +(-65*n^2+365*n-396)*a(n-3) -12*(3*n-5)*(3*n-7)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
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MATHEMATICA
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Table[Sum[(-1)^i*(i+1)*Binomial[3*n-2*i, n-i]/(2*n-i+1), {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2014 *)
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PROG
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(PARI) for(n=0, 50, print1(sum(k=0, n, (-1)^k*(k+1)*binomial(3*n-2*k, n-k)/(2*n - k+1)), ", ")) \\ G. C. Greubel, Feb 07 2017
(PARI) Vec((g->g/(1+x*g))(1 + serreverse(x/(1+x)^3 + O(x^25)))) \\ Andrew Howroyd, Nov 12 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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