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A066357
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Number of ordered (i.e. planar) trees on 2n nodes with every subtree at the root having an even number of edges.
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5
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1, 1, 6, 53, 554, 6362, 77580, 986253, 12927170, 173452334, 2370742868, 32892031042, 462030186916, 6557906929108, 93909078262808, 1355087936016957, 19684187540818866, 287612514032460070
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) is the number of walks from (0,0) to (2n,2n) using steps (0,1) and (1,0) which never stray below the line y=x and which avoid the points (m,m) m odd. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 14 2003
Series reversion of Sum_{n>0} -a(n)(-x)^n is g.f. of A005900.
a(n) = ((2^(4n))/Gamma(1/2)) * ((6*(2n+1)*Gamma(2n+1/2)/Gamma(2n+3))-2Gamma(n+1/2)/Gamma(n+2)) [From David Dickson (dcmd(AT)unimelb.edu.au), Nov 10 2009]
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REFERENCES
| D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (p. 333).
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LINKS
| C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
A. de Mier and M. Noy, A solution to the tennis ball problem
J.-G. Luque and J.-Y. Thibon, Noncommutative Symmetric Functions Associated with a Code, Lazard Elimination and Witt Vectors, to appear in Journal of Automata, Languages and Combinatorics.
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FORMULA
| For n>0, a(n)=sum(C(2*r-1)*a(n-r), r=1..n). Here C(2*r-1) is a Catalan number (A000108). - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 14 2003
G.f.: 2/(1+4sqrt(x)/(sqrt(1+4sqrt(x))-sqrt(1-4sqrt(x)))).
a(n)(2n-1)(n+1)n=a(n-1)(32*n^2-64*n+39)2n-a(n-2)(2n-3)(4n-5)(4n-7)16, n>1.
a(0)=1,a(n)=(1/n)*sum{k=0..n, C(4n,k)*C(3n-k-2,n-k-1)},n>1. - Paul Barry (pbarry(AT)wit.ie), Apr 09 2007
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MAPLE
| gf := (1-sqrt(1-4*z)-sqrt(1+4*z)+sqrt(1-16*z^2))/(z*(sqrt(1-4*z)-sqrt(1+4*z))):s := series(gf, z, 80): for i from 0 to 50 by 2 do printf(`%d, `, coeff(s, z, i)) od:
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PROG
| (PARI) a(n)=local(A); if(n<1, n==0, A=sqrt(1+4*x+O(x^(2*n+2))); A-=subst(A, x, -x); polcoeff(((2*A-8*x)/A^2)^2, 2*n))
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CROSSREFS
| Row sums of A078990. First column of A079513.
Sequence in context: A164894 A027835 A055973 * A185148 A109092 A068416
Adjacent sequences: A066354 A066355 A066356 * A066358 A066359 A066360
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KEYWORD
| nonn,easy
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AUTHOR
| Louis Shapiro (lshapiro(AT)howard.edu) Feb 01 2002
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EXTENSIONS
| More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Feb 11, 2002
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