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%I #16 Apr 09 2024 04:30:06
%S 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,
%T 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,
%U 56,0,21,0,6,0,1,462,42,210,42,84,28,28,14,7,5,1,1,924,0,462,0,210,0,84,0,28,0,7,0,1
%N Triangle read by rows, T(n,k) = number of symmetric ordered trees with n edges and root degree k (1 <= k <= n).
%C Number of symmetric Dyck n-paths with k returns to the x-axis. - _David Scambler_, Aug 16 2012
%F 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).
%F Sum_{k=1..n} T(n,k) = A001405(n).
%F T(n,1) = A001405(n-1).
%F Sum_{k=1..n} k*T(n,k) = A143360(n).
%F Sum_{k=2..n} T(n,k) = A037952(n). - _R. J. Mathar_, Sep 24 2021
%e Triangle starts:
%e 1;
%e 1, 1;
%e 2, 0, 1;
%e 3, 1, 1, 1;
%e 6, 0, 3, 0, 1;
%e 10, 2, 4, 2, 1, 1;
%e 20, 0, 10, 0, 4, 0, 1;
%e 35, 5, 15, 5, 5, 3, 1, 1;
%p 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
%t Module[{nmax = 13, G, C, S},
%t G = (1 + t*z*S[z])/(1 - t^2*z^2*C[z^2]) - 1;
%t S[z_] = 1/(1 - z - z^2*C[z^2]) ;
%t C[z_] = (1 - Sqrt[1 - 4 z])/(2 z);
%t CoefficientList[#/t + O[t]^nmax, t]& /@
%t CoefficientList[G/z + O[z]^nmax, z]
%t ] // Flatten (* _Jean-François Alcover_, Apr 09 2024 *)
%Y Cf. A001405, A000108 (column 2), A143360, A037952 (column 3).
%K nonn,tabl,changed
%O 1,4
%A _Emeric Deutsch_, Aug 15 2008
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