A075864 G.f. satisfies A(x) = 1 + Sum_{n>=0} (x*A(x))^(2^n).

1, 1, 2, 4, 10, 26, 72, 204, 594, 1762, 5318, 16270, 50360, 157392, 496016, 1574432, 5028962, 16152194, 52133154, 169004450, 550036778, 1796512970, 5886709502, 19346204982, 63751851400, 210605429496, 697337388556, 2313871053172, 7692939444640, 25623793107344

Offset

0,3

Comments

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

Links

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

Formula

G.f. A(x) satisfies x*A(x) = series_reversion( x / ( 1 + Sum_{k>=0} x^(2^k) ) ). - Joerg Arndt, Apr 01 2019

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, y-2^j, false), j=0..ilog2(y)), 0)))
    end:
a:= n-> b(2*n, 0, true):
seq(a(n), n=0..32);  # Alois P. Heinz, Apr 01 2019

Mathematica

seq = {};
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, d], If[d == 0, 1, 2*d]]]];
For[n = 0, n <= 27, n++, seq = Append[seq, f[n, n, 0]]]; seq
(* David Scambler, May 07 2012 *)
A[_] = 0; m = 32;
Do[A[x_] = 1+Sum[(x A[x])^(2^n)+O[x]^m, {n, 0, Log[2, m]//Ceiling}], {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, May 20 2022 *)

Prog

(PARI) N=66; K=ceil(log(N)/log(2))+1; x='x+O('x^N); Vec(serreverse(x/(1 + sum(k=0, K, x^(2^k) ) ) ) ) - Joerg Arndt, Apr 01 2019

Crossrefs

Cf. A075853.

Sequence in context: A149813 A149814 A125108 * A180023 A154835 A049145

Adjacent sequences: A075861 A075862 A075863 * A075865 A075866 A075867

Keyword

nonn

Author

Paul D. Hanna, Oct 15 2002

Status

approved