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 A212382 Number A(n,k) of Dyck n-paths all of whose ascents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 14, 1, 1, 1, 1, 1, 5, 42, 1, 1, 1, 1, 1, 2, 12, 132, 1, 1, 1, 1, 1, 1, 6, 30, 429, 1, 1, 1, 1, 1, 1, 2, 16, 79, 1430, 1, 1, 1, 1, 1, 1, 1, 7, 37, 213, 4862, 1, 1, 1, 1, 1, 1, 1, 2, 22, 83, 584, 16796, 1 (list; table; graph; refs; listen; history; text; internal format)
 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 `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

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Last modified August 25 21:00 EDT 2019. Contains 326324 sequences. (Running on oeis4.)