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

From Paul D. Hanna, Jul 12 2024: (Start)

G.f. A(x) = x*Sum_{n>=0} a(n)*x^n (offset 1) satisfies the following formulas.

(1) A(x)^2 = A( x*A(x)/(1-x) ).

(2) A(x)^4 = A( x*A(x)^3/(1 - x - x*A(x)) ).

(3) A(x)^8 = A( x*A(x)^7/(1 - x - x*A(x) - x*A(x)^3) ).

(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) / (1 - x*Sum_{k=0..n-1} A(x)^(2^k-1)) ) for n >= 1.

The radius of convergence r and A(r) satisfy r = 1/(Sum_{n>=0} 2^n*A(r)^(2^n-1)) and A(r) = A( A(r)*r/(1-r) )^(1/2), where r = 0.285128929740568796881205193649402054331317007180873... and A(r) = 0.621954965556741102287309027445345554104820417676869...

(End)

Example

G.f. (offset 0): A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 72*x^6 + 204*x^7 + 594*x^8 + 1762*x^9 + 5318*x^10 + ...
SPECIFIC VALUES.
The following values are for the g.f. at offset 1: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 26*x^6 + 72*x^7 + 204*x^8 + ...
A(t) = 1/2 at t = 0.275266504782383938866239471561026684712237255315...
where 1/4 = A( (1/2)*t/(1-t) ) and t = (1/2)/(1 + Sum_{n>=0} 1/2^(2^n)).
A(t) = 1/3 at t = 0.228789618697442059759075468255467011039543924763...
where 1/9 = A( (1/3)*t/(1-t) ) and t = (1/3)/(1 + Sum_{n>=0} 1/3^(2^n)).
A(1/4) = 0.392935121163880589695619242847181861583875787578...
where A(1/4)^2 = A( (1/3)*A(1/4) ).
A(1/5) = 0.269480257065638376643289111191173593741789085897...
where A(1/5)^2 = A( (1/4)*A(1/5) ).
A(1/6) = 0.209130395397987995845331540196686970439063098884...
where A(1/6)^2 = A( (1/5)*A(1/6) ).

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
(PARI) {a(n) = my(A=[1], Ax);
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^2 - subst(Ax, x, x*Ax/(1-x) ), #A+1) ); A[n]}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 12 2024

Crossrefs

Cf. A075853, A374565.

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