OFFSET
0,9
COMMENTS
Lengths of descents are unrestricted.
For p>0 is column p asymptotic to a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where r and s are real roots (0 < r < 1) of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
Vaclav Kotesovec, Asymptotic of subsequences of A212382
FORMULA
G.f. of column k>0 satisfies: A_k(x) = 1+x*A_k(x)/(1-(x*A_k(x))^k), g.f. of column k=0: A_0(x) = 1/(1-x).
G.f. of column k>0 is series_reversion(B(x))/x where B(x) = x/(1 + x + x^(k+1) + x^(2*k+1) + x^(3*k+1) + ... ) = x/(1+x/(1-x^k)); for Dyck paths with allowed ascent lengths {u_1, u_2, ...} use B(x) = x/( 1 + sum(k>=1, x^{u_k} ) ). - Joerg Arndt, Apr 23 2016
EXAMPLE
A(0,k) = 1: the empty path.
A(3,0) = 1: UDUDUD.
A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.
A(3,2) = 2: UDUDUD, UUUDDD.
A(5,3) = 6: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, UUUUDUDDDD.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 1, 1, 1, 1, 1, ...
1, 5, 2, 1, 1, 1, 1, 1, ...
1, 14, 5, 2, 1, 1, 1, 1, ...
1, 42, 12, 6, 2, 1, 1, 1, ...
1, 132, 30, 16, 7, 2, 1, 1, ...
1, 429, 79, 37, 22, 8, 2, 1, ...
MAPLE
b:= proc(x, y, k, u) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, k, true)+
`if`(u, add(b(x-(k*t+1), y, k, false), t=0..(x-1)/k), 0)))
end:
A:= (n, k)-> `if`(k=0, 1, b(n, n, k, true)):
seq(seq(A(n, d-n), n=0..d), d=0..15);
# second Maple program
A:= (n, k)-> `if`(k=0, 1, coeff(series(RootOf(
A||k=1+x*A||k/(1-(x*A||k)^k), A||k), x, n+1), x, n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);
MATHEMATICA
b[x_, y_, k_, u_] := b[x, y, k, u] = If[x<0 || y<x, 0, If[x == 0 && y == 0, 1, b[x, y-1, k, True] + If[u, Sum[b[x-(k*t+1), y, k, False], {t, 0, (x-1)/k}], 0]]]; A[n_, k_] := If[k == 0, 1, b[n, n, k, True]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from first Maple program *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 12 2012
STATUS
approved