A134527 G.f. A(x) satisfies: A(x) = Sum_{n>=0} [x*A(x)]^(2^n-1).

1, 1, 1, 2, 5, 11, 24, 58, 149, 385, 1001, 2652, 7140, 19384, 52944, 145590, 402949, 1121117, 3133255, 8793372, 24774557, 70045871, 198672464, 565144064, 1611946284, 4609140916, 13209415116, 37937455636, 109171460104, 314736939884, 908930799572, 2629120466966

Offset: 0

Author

Paul D. Hanna, Nov 19 2007

Keywords

• nonn

Comments

Number of Dyck n-paths with all ascent lengths being 1 less than a power of 2. [David Scambler, May 07 2012]

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A075864.

Programs

Maple

b:= proc(x, y, t) option remember; `if`(x<0 or y>x, 0,
      `if`(x=0, 1, b(x-1, y+1, true)+`if`(t, add(
       b(x-2^j+1, y-2^j+1, false), j=1..ilog2(y+1)), 0)))
    end:
a:= n-> b(2*n, 0, true):
seq(a(n), n=0..32);  # Alois P. Heinz, Apr 01 2019

Mathematica

f[x_, y_, d_] := f[x, y, d] = If[x < 0 || y < x, 0, If[x == 0 && y == 0, 1, f[x - 1, y, 0] + f[x, y - If[d == 0, 1, 2*d], If[d == 0, 1, 2*d]]]];Table[f[n, n, 0], {n, 0, 28}] (* David Scambler, May 07 2012 *)

PARI

a(n)=polcoeff(serreverse(x/sum(j=0,#binary(n),x^(2^j-1)+ x*O(x^n))),n)

Formula

G.f.: A(x) = Series_Reversion{x/[Sum_{n>=0} x^(2^n-1)]}.