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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).

%I A143359 #16 Apr 09 2024 04:30:06
%S A143359 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 A143359 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 A143359 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 A143359 Triangle read by rows, T(n,k) = number of symmetric ordered trees with n edges and root degree k (1 <= k <= n).
%C A143359 Number of symmetric Dyck n-paths with k returns to the x-axis. - _David Scambler_, Aug 16 2012
%F A143359 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 A143359 Sum_{k=1..n} T(n,k) = A001405(n).
%F A143359 T(n,1) = A001405(n-1).
%F A143359 Sum_{k=1..n} k*T(n,k) = A143360(n).
%F A143359 Sum_{k=2..n} T(n,k) = A037952(n). - _R. J. Mathar_, Sep 24 2021
%e A143359 Triangle starts:
%e A143359    1;
%e A143359    1,  1;
%e A143359    2,  0,  1;
%e A143359    3,  1,  1,  1;
%e A143359    6,  0,  3,  0,  1;
%e A143359   10,  2,  4,  2,  1,  1;
%e A143359   20,  0, 10,  0,  4,  0,  1;
%e A143359   35,  5, 15,  5,  5,  3,  1,  1;
%p A143359 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 A143359 Module[{nmax = 13, G, C, S},
%t A143359    G = (1 + t*z*S[z])/(1 - t^2*z^2*C[z^2]) - 1;
%t A143359    S[z_] = 1/(1 - z - z^2*C[z^2]) ;
%t A143359    C[z_] = (1 - Sqrt[1 - 4 z])/(2 z);
%t A143359    CoefficientList[#/t + O[t]^nmax, t]& /@
%t A143359    CoefficientList[G/z + O[z]^nmax, z]
%t A143359 ] // Flatten (* _Jean-François Alcover_, Apr 09 2024 *)
%Y A143359 Cf. A001405, A000108 (column 2), A143360, A037952 (column 3).
%K A143359 nonn,tabl
%O A143359 1,4
%A A143359 _Emeric Deutsch_, Aug 15 2008