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%I #27 May 20 2022 12:59:57
%S 1,1,2,4,10,26,72,204,594,1762,5318,16270,50360,157392,496016,1574432,
%T 5028962,16152194,52133154,169004450,550036778,1796512970,5886709502,
%U 19346204982,63751851400,210605429496,697337388556,2313871053172,7692939444640,25623793107344
%N G.f. satisfies A(x) = 1 + Sum_{n>=0} (x*A(x))^(2^n).
%C Number of Dyck n-paths with all ascent lengths being a power of 2. [_David Scambler_, May 07 2012]
%H Alois P. Heinz, <a href="/A075864/b075864.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f. A(x) satisfies x*A(x) = series_reversion( x / ( 1 + Sum_{k>=0} x^(2^k) ) ). - _Joerg Arndt_, Apr 01 2019
%p b:= proc(x, y, t) option remember; `if`(x<0 or y>x, 0,
%p `if`(x=0, 1, b(x-1, y+1, true)+`if`(t, add(
%p b(x-2^j, y-2^j, false), j=0..ilog2(y)), 0)))
%p end:
%p a:= n-> b(2*n, 0, true):
%p seq(a(n), n=0..32); # _Alois P. Heinz_, Apr 01 2019
%t seq = {};
%t f[x_, y_, d_] :=
%t f[x, y, d] =
%t If[x < 0 || y < x , 0,
%t If[x == 0 && y == 0, 1,
%t f[x - 1, y, 0] + f[x, y - If[d == 0, 1, d], If[d == 0, 1, 2*d]]]];
%t For[n = 0, n <= 27, n++, seq = Append[seq, f[n, n, 0]]]; seq
%t (* _David Scambler_, May 07 2012 *)
%t A[_] = 0; m = 32;
%t Do[A[x_] = 1+Sum[(x A[x])^(2^n)+O[x]^m, {n, 0, Log[2, m]//Ceiling}], {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, May 20 2022 *)
%o (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
%Y Cf. A075853.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 15 2002
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