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A281688 Expansion of Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).

%I A281688 #9 May 04 2021 18:39:39
%S A281688 1,3,5,10,14,23,29,45,55,79,93,130,150,199,225,296,332,423,469,594,
%T A281688 654,807,881,1085,1179,1423,1537,1850,1990,2355,2521,2983,3185,3719,
%U A281688 3957,4618,4902,5655,5985,6909,7299,8343,8793,10050,10574,11979,12577,14260,14952,16823,17609,19818,20718,23155,24169,27033
%N A281688 Expansion of Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).
%C A281688 Total number of parts in all partitions of n into powers of 2 (A000079).
%C A281688 Convolution of A001511 and A018819.
%H A281688 Alois P. Heinz, <a href="/A281688/b281688.txt">Table of n, a(n) for n = 1..20000</a>
%H A281688 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A281688 G.f.: Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).
%e A281688 a(4) = 10 because we have [4], [2, 2], [2, 1, 1], [1, 1, 1, 1] and 1 + 2 + 3 + 4 = 10.
%p A281688 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p->
%p A281688       `if`(p>n, 0, (h-> h+[0, h[1]])(b(n-p, i))))(2^i)+b(n, i-1)))
%p A281688     end:
%p A281688 a:= n-> b(n, ilog2(n))[2]:
%p A281688 seq(a(n), n=1..56);  # _Alois P. Heinz_, May 04 2021
%t A281688 Rest[CoefficientList[Series[Sum[x^2^i/(1 - x^2^i), {i, 0, 20}]/Product[1 - x^2^j, {j, 0, 20}], {x, 0, 56}], x]]
%Y A281688 Cf. A000079, A001511, A018819, A343944.
%K A281688 nonn
%O A281688 1,2
%A A281688 _Ilya Gutkovskiy_, Jan 27 2017