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A030983
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Number of rooted noncrossing trees with n nodes such that root has degree 1 and the child of the root has degree at least 2.
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4
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0, 3, 16, 83, 442, 2420, 13566, 77539, 450340, 2650635, 15777450, 94815732, 574518536, 3506232184, 21533144486, 132980242755, 825304177544, 5144743785545, 32199189658020, 202252227085755, 1274578959894450, 8056409137803600, 51063344718826440
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OFFSET
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3,2
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COMMENTS
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With offset 0 (i.e., a(0) = 0 and a(1) = 3), a(n) is the total number of down-steps after the final up-step in all 2_1-Dyck paths of length 3*n.
A 2_1-Dyck path is a lattice path with steps U = (1, 2) and d = (1, -1) that starts at (0,0), stays (weakly) above the line y = -1, and ends at the x-axis.
For n = 2, a(2) = 16 is the total number of down-steps after the final up-step in dUddUd, dUdUdd, dUUddd, UdddUd, UddUdd, UdUddd, UUdddd (thus, 1 + 2 + 3 + 1 + 2 + 3 + 4). (End)
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LINKS
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FORMULA
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a(n) = (19*n - 31)*binomial(3*n - 8, n - 4)/(n - 1)/(2*n - 3).
a(n) = binomial(3*n-5, 2*n-3)/(n-1) - 2*binomial(3*n-8, 2*n-5)/(n-2), n > 2.
a(n) = Sum_{i=1..n-3} binomial(3*i-2, 2*i-1) * binomial(3*(n-i-2), 2*(n-i-2)-1)/ (i*(n-i-2)). (End)
a(n) ~ (76*3^(3*n - 15/2))/(4^n*sqrt(Pi)*n^(3/2)). - Peter Luschny, Aug 08 2020
D-finite with recurrence 2*(n-1)*(2*n-3)*a(n) +(-43*n^2+196*n-213)*a(n-1) +2*(62*n^2-446*n+759)*a(n-2) -12*(3*n-14)*(3*n-16)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
D-finite with recurrence 2*(n-1)*(n-4)*(2*n-3)*(19*n-50)*a(n) -3*(3*n-10)*(3*n-8)*(n-3)*(19*n-31)*a(n-1)=0. - R. J. Mathar, Jul 26 2022
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MAPLE
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h := arcsin((3*sqrt(3)*sqrt(x))/2)/3:
gf := x*(64/9)*sin(h)^6*(1 - sin(h)^2*(8/9)): ser := series(gf, x, 32):
# Recurrence:
a := proc(n) option remember; if n < 4 then return 0 fi; if n = 4 then return 3 fi;
-((378*n^3 - 4536*n^2 + 18102*n - 24024)*a(n - 2) + (-1271*n^3 + 10308*n^2 - 26857*n + 22020)*a(n - 1))/(180*n^3 - 1170*n^2 + 2070*n - 1080) end:
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MATHEMATICA
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a[n_] := Binomial[3n-5, n-2]/(n-1) - 2 Binomial[3n-8, n-3]/(n-2);
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PROG
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(PARI) a(n)=(19*n-31)*binomial(3*n-8, n-4)/(n-1)/(2*n-3); /* Joerg Arndt, Mar 07 2013 */
(PARI) concat(0, Vec((g->g^3*(3-2*g))(serreverse(x-2*x^2+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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STATUS
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approved
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