OFFSET
1,4
COMMENTS
Number of symmetric Dyck n-paths with k returns to the x-axis. - David Scambler, Aug 16 2012
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
MATHEMATICA
Module[{nmax = 13, G, C, S},
G = (1 + t*z*S[z])/(1 - t^2*z^2*C[z^2]) - 1;
S[z_] = 1/(1 - z - z^2*C[z^2]) ;
C[z_] = (1 - Sqrt[1 - 4 z])/(2 z);
CoefficientList[#/t + O[t]^nmax, t]& /@
CoefficientList[G/z + O[z]^nmax, z]
] // Flatten (* Jean-François Alcover, Apr 09 2024 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Aug 15 2008
STATUS
approved