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A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).

%I A082392 #32 Sep 15 2023 05:25:34
%S A082392 1,1,2,1,4,2,8,1,16,4,32,2,64,8,128,1,256,16,512,4,1024,32,2048,2,
%T A082392 4096,64,8192,8,16384,128,32768,1,65536,256,131072,16,262144,512,
%U A082392 524288,4,1048576,1024,2097152,32,4194304,2048,8388608,2,16777216
%N A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).
%H A082392 Seiichi Manyama, <a href="/A082392/b082392.txt">Table of n, a(n) for n = 0..5000</a>
%H A082392 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A082392 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>
%F A082392 a(0) = 1, a(2*n) = 2^n, a(2*n+1) = a(n).
%F A082392 a(n) = 2^A025480(n) = 2^(A003602(n)-1).
%F A082392 a((2*n+1)*2^p-1) = 2^n, p >= 0 and n >= 0. - _Johannes W. Meijer_, Feb 11 2013
%p A082392 nmax := 48: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := 2^n od: od: seq(a(n), n=0..nmax); # _Johannes W. Meijer_, Feb 11 2013
%p A082392 A082392 := proc(n)
%p A082392     2^A025480(n) ;
%p A082392 end proc:
%p A082392 seq(A082392(n),n=0..100) ; # _R. J. Mathar_, Jul 16 2020
%t A082392 a[n_] := 2^(((n+1)/2^IntegerExponent[n+1, 2]+1)/2-1);
%t A082392 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Sep 15 2023 *)
%o A082392 (PARI) for(n=0, 50, l=ceil(log(n+1)/log(2)); t=polcoeff(sum(k=0, l, (x^2^k)/(1-2*x^2^(k+1)))/x + O(x^(n+1)), n); print1(t", ");) ;
%Y A082392 Cf. A045654, A220466.
%K A082392 nonn,easy
%O A082392 0,3
%A A082392 _Ralf Stephan_, Jun 07 2003