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

%I A075864 #43 Jul 23 2024 02:22:31
%S A075864 1,1,2,4,10,26,72,204,594,1762,5318,16270,50360,157392,496016,1574432,
%T A075864 5028962,16152194,52133154,169004450,550036778,1796512970,5886709502,
%U A075864 19346204982,63751851400,210605429496,697337388556,2313871053172,7692939444640,25623793107344
%N A075864 G.f. satisfies A(x) = 1 + Sum_{n>=0} (x*A(x))^(2^n).
%C A075864 Number of Dyck n-paths with all ascent lengths being a power of 2. - _David Scambler_, May 07 2012
%H A075864 Alois P. Heinz, <a href="/A075864/b075864.txt">Table of n, a(n) for n = 0..1000</a>
%F A075864 G.f. A(x) satisfies x*A(x) = series_reversion( x / ( 1 + Sum_{k>=0} x^(2^k) ) ). - _Joerg Arndt_, Apr 01 2019
%F A075864 From _Paul D. Hanna_, Jul 12 2024: (Start)
%F A075864 G.f. A(x) = x*Sum_{n>=0} a(n)*x^n (offset 1) satisfies the following formulas.
%F A075864 (1) A(x)^2 = A( x*A(x)/(1-x) ).
%F A075864 (2) A(x)^4 = A( x*A(x)^3/(1 - x - x*A(x)) ).
%F A075864 (3) A(x)^8 = A( x*A(x)^7/(1 - x - x*A(x) - x*A(x)^3) ).
%F A075864 (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.
%F A075864 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...
%F A075864 (End)
%e A075864 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 + ...
%e A075864 SPECIFIC VALUES.
%e A075864 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 + ...
%e A075864 A(t) = 1/2 at t = 0.275266504782383938866239471561026684712237255315...
%e A075864 where 1/4 = A( (1/2)*t/(1-t) ) and t = (1/2)/(1 + Sum_{n>=0} 1/2^(2^n)).
%e A075864 A(t) = 1/3 at t = 0.228789618697442059759075468255467011039543924763...
%e A075864 where 1/9 = A( (1/3)*t/(1-t) ) and t = (1/3)/(1 + Sum_{n>=0} 1/3^(2^n)).
%e A075864 A(1/4) = 0.392935121163880589695619242847181861583875787578...
%e A075864 where A(1/4)^2 = A( (1/3)*A(1/4) ).
%e A075864 A(1/5) = 0.269480257065638376643289111191173593741789085897...
%e A075864 where A(1/5)^2 = A( (1/4)*A(1/5) ).
%e A075864 A(1/6) = 0.209130395397987995845331540196686970439063098884...
%e A075864 where A(1/6)^2 = A( (1/5)*A(1/6) ).
%p A075864 b:= proc(x, y, t) option remember; `if`(x<0 or y>x, 0,
%p A075864       `if`(x=0, 1, b(x-1, y+1, true)+`if`(t, add(
%p A075864        b(x-2^j, y-2^j, false), j=0..ilog2(y)), 0)))
%p A075864     end:
%p A075864 a:= n-> b(2*n, 0, true):
%p A075864 seq(a(n), n=0..32);  # _Alois P. Heinz_, Apr 01 2019
%t A075864 seq = {};
%t A075864 f[x_, y_, d_] :=
%t A075864   f[x, y, d] =
%t A075864    If[x < 0 || y < x , 0,
%t A075864     If[x == 0 && y == 0, 1,
%t A075864      f[x - 1, y, 0] + f[x, y - If[d == 0, 1, d], If[d == 0, 1, 2*d]]]];
%t A075864 For[n = 0, n <= 27, n++, seq = Append[seq, f[n, n, 0]]]; seq
%t A075864 (* _David Scambler_, May 07 2012 *)
%t A075864 A[_] = 0; m = 32;
%t A075864 Do[A[x_] = 1+Sum[(x A[x])^(2^n)+O[x]^m, {n, 0, Log[2, m]//Ceiling}], {m}];
%t A075864 CoefficientList[A[x], x] (* _Jean-François Alcover_, May 20 2022 *)
%o A075864 (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
%o A075864 (PARI) {a(n) = my(A=[1], Ax);
%o A075864 for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
%o A075864 A[#A] = -polcoeff( Ax^2 - subst(Ax,x, x*Ax/(1-x) ), #A+1) ); A[n]}
%o A075864 for(n=1, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jul 12 2024
%Y A075864 Cf. A075853, A374565.
%K A075864 nonn
%O A075864 0,3
%A A075864 _Paul D. Hanna_, Oct 15 2002