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Total sum of parts which are powers of 2 in all partitions of n.
1

%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