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A143359
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Triangle read by rows, T(n,k) = number of symmetric ordered trees with n edges and root degree k (1 <= k <= n).
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2
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1, 1, 1, 2, 0, 1, 3, 1, 1, 1, 6, 0, 3, 0, 1, 10, 2, 4, 2, 1, 1, 20, 0, 10, 0, 4, 0, 1, 35, 5, 15, 5, 5, 3, 1, 1, 70, 0, 35, 0, 15, 0, 5, 0, 1, 126, 14, 56, 14, 21, 9, 6, 4, 1, 1, 252, 0, 126, 0, 56, 0, 21, 0, 6, 0, 1, 462, 42, 210, 42, 84, 28, 28, 14, 7, 5, 1, 1, 924, 0, 462, 0, 210, 0, 84, 0, 28
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OFFSET
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1,4
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COMMENTS
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Number of symmetric Dyck n-paths with k returns to the x-axis. - David Scambler, Aug 16 2012
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LINKS
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Table of n, a(n) for n=1..87.
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FORMULA
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G.f. = (1+t*z*S)/(1-t^2*z^2*C(z^2))-1, where S = 1/(1-z-z^2*C(z^2)) is the g.f. of the sequence binomial(n, floor(n/2)) (A001405) and C(z) = (1-sqrt(1-4z))/(2z) is the generating function of the Catalan numbers (A000108).
Sum_{k=1..n} T(n,k) = A001405(n).
T(n,1) = A001405(n-1).
Sum_{k=1..n} k*T(n,k) = A143360(n).
Sum_{k=2..n} T(n,k) = A037952(n). - R. J. Mathar, Sep 24 2021
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EXAMPLE
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Triangle starts:
1;
1, 1;
2, 0, 1;
3, 1, 1, 1;
6, 0, 3, 0, 1;
10, 2, 4, 2, 1, 1;
20, 0, 10, 0, 4, 0, 1;
35, 5, 15, 5, 5, 3, 1, 1;
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MAPLE
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C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: S:=1/(1-z-z^2*C(z^2)): G:=(1+t*z*S)/(1-t^2*z^2*C(z^2))-1: Gser:=simplify(series(G, z=0, 15)): for n to 13 do P[n]:=coeff(Gser, z, n) end do: for n to 13 do seq(coeff(P[n], t, j), j=1..n) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A001405, A000108 (column 2), A143360, A037952 (column 3).
Sequence in context: A094184 A078805 A122837 * A291316 A130504 A044942
Adjacent sequences: A143356 A143357 A143358 * A143360 A143361 A143362
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Aug 15 2008
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STATUS
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approved
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