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 A143359 Triangle read by rows, T(n,k) = number of symmetric ordered trees with n edges and root degree k (1 <= k <= n). 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Number of symmetric Dyck n-paths with k returns to the x-axis. - David Scambler, Aug 16 2012 LINKS FORMULA 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 EXAMPLE 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; MAPLE 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 CROSSREFS 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 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Aug 15 2008 STATUS approved

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Last modified March 24 19:07 EDT 2023. Contains 361510 sequences. (Running on oeis4.)