OFFSET
0,13
COMMENTS
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see p. 236, Exercise 6.34 d.
LINKS
Alois P. Heinz, Rows n = 0..32, flattened
FORMULA
The generating polynomial for row n is P[n](t) = ((1+t)/(1+t^n))*binomial[2n,n]/[n+1], where [n+1]=1+t+t^2+...+t^n and binomial[2n,n] is a Gaussian polynomial.
EXAMPLE
T(4,5)=2 because we have UD.UUD.UDD (2+5-2=5) and UUUDDD.UD (6-1=5); here U=(1,1), D=(1,-1) and each valley is shown by a dot.
Triangle starts:
1;
1;
1,1;
1,1,1,1,1;
1,1,1,2,2,2,2,1,1,1;
1,1,1,2,3,3,4,4,4,4,4,3,3,2,1,1,1;
MAPLE
br:=proc(n) options operator, arrow: sum(q^i, i=0..n-1) end proc: f:=proc(n) options operator, arrow: product(br(j), j=1..n) end proc: cbr:=proc(n, k) options operator, arrow: f(n)/(f(k)*f(n-k)) end proc: P:=proc(n) options operator, arrow: sort(expand(simplify((q+1)*cbr(2*n, n)/(br(n+1)*(1+q^n))))) end proc: 1; for n to 7 do seq(coeff(P(n), q, k), k=0..(n-1)^2) end do; # yields sequence in triangular form
MATHEMATICA
g[k_] := (1 - t^k)/(1 - t);
gpol[n_, k_] := If[0 <= k <= n, Product[g[n - j + 1]/g[j], {j, 1, k}], 0];
P[n_] := ((1 + t)/(1 + t^n)) gpol[2n, n]/Sum[t^k, {k, 0, n}];
T[n_] := CoefficientList[P[n] + O[t]^(n^2), t]; T[0] = {1};
T /@ Range[0, 7] // Flatten (* Jean-François Alcover, Feb 16 2021 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Nov 06 2008
STATUS
approved