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A127531
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Number of jumps in all binary trees with n edges.
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2
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0, 0, 2, 13, 64, 285, 1210, 5005, 20384, 82212, 329460, 1314610, 5230016, 20764055, 82317690, 326012925, 1290244800, 5103910680, 20183646780, 79802261190, 315492902400, 1247247742650, 4930910180196, 19495286167698, 77085553829824
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OFFSET
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1,3
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COMMENTS
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In the preorder traversal of a binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump.
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LINKS
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FORMULA
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a(n) = binomial(2*n, n-2) - binomial(2*n - 2, n-2).
a(n) = 4^(n-1)*(n-1)*(3*n^2-5*n-2)*Gamma(n-1/2)/(sqrt(Pi)*Gamma(n+3)).
a(n) ~ 4^n*(3-139/(8*n)+8595/(128*n^2)-234745/(1024*n^3)+24282657/(32768*n^4)) /sqrt(n*Pi). (End)
D-finite with recurrence -5*(n+2)*(n-3)*a(n) +(19*n^2-26*n-5)*a(n-1) +2*(n-2)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Jul 26 2022
D-finite with recurrence +(n-3)*(3*n-2)*(n+2)*a(n) -2*(n-1)*(3*n+1)*(2*n-3)*a(n-1)=0. - R. J. Mathar, Jul 26 2022
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MAPLE
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seq(binomial(2*n, n-2)-binomial(2*n-2, n-2), n=1..28);
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MATHEMATICA
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Table[Binomial[2n, n-2] - Binomial[2n-2, n-2], {n, 30}] (* or *) Table[4^(n-1)(n-1)(3n^2 -5n-2)Gamma[n-1/2]/(Sqrt[Pi]Gamma[n+3]), {n, 30}] (* Michael De Vlieger, Dec 19 2015 *)
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PROG
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(Magma) [Binomial(2*n, n-2) - Binomial(2*n - 2, n-2): n in [1..30]]; // Vincenzo Librandi, Dec 20 2015
(PARI) vector(30, n, binomial(2*n, n-2) -binomial(2*n-2, n-2) ) \\ G. C. Greubel, Mar 19 2017
(Magma) [Binomial(2*n, n-2) -Binomial(2*n-2, n-2): n in [1..30]]; // G. C. Greubel, May 08 2019
(Sage) [binomial(2*n, n-2) -binomial(2*n-2, n-2) for n in (1..30)] # G. C. Greubel, May 08 2019
(GAP) List([1..30], n-> Binomial(2*n, n-2) -Binomial(2*n-2, n-2)) # G. C. Greubel, May 08 2019
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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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