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%I #5 Mar 07 2021 03:56:15
%S 0,1,4,6,17,24,43,64,115,159,247,347,513,704,1001,1350,1894,2513,3408,
%T 4489,5989,7786,10226,13172,17079,21800,27938,35362,44900,56402,70959,
%U 88545,110617,137108,170051,209599,258328,316685,388072,473331,577026,700524,849775,1027167
%N Total sum of parts which are powers of 2 in all partitions of n.
%F G.f.: Sum_{k>=0} 2^k*x^(2^k)/(1 - x^(2^k)) / Product_{j>=1} (1 - x^j).
%F a(n) = Sum_{k=1..n} A038712(k) * A000041(n-k).
%e For n = 4 we have:
%e ------------------------------------
%e Partitions Sum of parts
%e . which are powers of 2
%e ------------------------------------
%e 4 ..................... 4
%e 3 + 1 ................. 1
%e 2 + 2 ................. 4
%e 2 + 1 + 1 ............. 4
%e 1 + 1 + 1 + 1 ......... 4
%e ------------------------------------
%e Total ................ 17
%e So a(4) = 17.
%t nmax = 43; CoefficientList[Series[Sum[2^k x^(2^k)/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
%t Table[Sum[(2^IntegerExponent[2 k, 2] - 1) PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]
%Y Cf. A000041, A000079, A038712, A066186, A342230.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Mar 06 2021
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