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A281811 Expansion of Sum_{i>=0} x^(2^i) / (1 - Sum_{j>=0} x^(2^j))^2.

%I A281811 #11 Aug 07 2019 18:04:44
%S A281811 1,3,7,16,34,71,143,286,562,1096,2114,4054,7720,14631,27591,51834,
%T A281811 97018,181030,336810,625062,1157288,2138200,3942858,7257830,13338024,
%U A281811 24474978,44848232,82073852,150016328,273893503,499534495,910161570,1656786466,3013237398,5475710770,9942780954,18040712384,32711070838
%N A281811 Expansion of Sum_{i>=0} x^(2^i) / (1 - Sum_{j>=0} x^(2^j))^2.
%C A281811 Total number of parts in all compositions (ordered partitions) of n into powers of 2 (A000079).
%H A281811 Vaclav Kotesovec, <a href="/A281811/b281811.txt">Table of n, a(n) for n = 1..4000</a>
%H A281811 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A281811 G.f.: Sum_{i>=0} x^(2^i) / (1 - Sum_{j>=0} x^(2^j))^2.
%F A281811 a(n) ~ c * n / r^n, where r = 0.566123792684559918241681653033264449147... is the root of the equation Sum_{j>=0} r^(2^j) = 1 and c = 0.34432689951558638915900387175922521737229978512101795819134... . - _Vaclav Kotesovec_, Feb 17 2017
%e A281811 a(4) = 16 because we have [4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 1 + 2 + 3 + 3 + 3 + 4 = 16.
%p A281811 b:= proc(n) option remember; `if`(n=0, [1, 0], add(
%p A281811       (p-> p+[0, p[1]])(b(n-2^j)), j=0..ilog2(n)))
%p A281811     end:
%p A281811 a:= n-> b(n)[2]:
%p A281811 seq(a(n), n=1..55);  # _Alois P. Heinz_, Aug 07 2019
%t A281811 nmax = 38; Rest[CoefficientList[Series[Sum[x^2^i, {i, 0, nmax}]/(1 - Sum[x^2^j, {j, 0, nmax}])^2, {x, 0, nmax}], x]]
%t A281811 nmax = 40; Rest[CoefficientList[Series[Sum[x^(2^i), {i, 0, Floor[Log[nmax]/Log[2]] + 1}]/(1 - Sum[x^(2^j), {j, 0, Floor[Log[nmax]/Log[2]] + 1}])^2, {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Feb 17 2017 *)
%Y A281811 Cf. A000079, A023359.
%K A281811 nonn
%O A281811 1,2
%A A281811 _Ilya Gutkovskiy_, Jan 30 2017