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A162551
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a(n) = 2 * C(2*n,n-1).
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16
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0, 2, 8, 30, 112, 420, 1584, 6006, 22880, 87516, 335920, 1293292, 4992288, 19315400, 74884320, 290845350, 1131445440, 4407922860, 17194993200, 67156001220, 262564816800, 1027583214840, 4025232800160, 15780742227900, 61915399071552
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OFFSET
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0,2
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COMMENTS
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Total length of all Dyck paths of length 2n.
a(n) equals the diagonal element A(n,n) of matrix A whose element A(i,j) = A(i-1,j) + A(i,j-1). - Carmine Suriano, May 10 2010
a(n) is also the number of solid (3 dimensions) standard Young tableaux of shape [[n,n],[1]]. - Thotsaporn Thanatipanonda, Feb 27 2012
With offset = 1, a(n) is the total number of nodes over all binary trees with one child internal and one child external. - Geoffrey Critzer, Feb 23 2013
a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off the diagonal y = x an odd number of times. Details can be found in Section 4.2 in Pan and Remmel's link. - Ran Pan, Feb 01 2016
a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that cross the diagonal y = x an odd number of times. Details can be found in Section 4.3 in Pan and Remmel's link. - Ran Pan, Feb 01 2016
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REFERENCES
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R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley 1996, page 141.
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LINKS
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FORMULA
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E.g.f.: exp(2*x)*2*(BesselI(1,2*x)). - Peter Luschny, Aug 26 2012
E.g.f.: 2*Q(0) - 2, where Q(k) = 1 - 2*x/(k + 1 - (k + 1)*(2*k + 3)/(2*k + 3 - (k + 2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
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MATHEMATICA
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nn=25; Drop[CoefficientList[Series[(1-2x)/(1-4x)^(1/2), {x, 0, nn}], x], 1] (* Geoffrey Critzer, Feb 23 2013 *)
Table[2Binomial[2n, n-1], {n, 0, 30}] (* Harvey P. Dale, Oct 26 2016 *)
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PROG
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(Haskell)
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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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