cp's OEIS Frontend

A342231 Total sum of parts which are powers of 2 in all partitions of n.

%I A342231 #5 Mar 07 2021 03:56:15
%S A342231 0,1,4,6,17,24,43,64,115,159,247,347,513,704,1001,1350,1894,2513,3408,
%T A342231 4489,5989,7786,10226,13172,17079,21800,27938,35362,44900,56402,70959,
%U A342231 88545,110617,137108,170051,209599,258328,316685,388072,473331,577026,700524,849775,1027167
%N A342231 Total sum of parts which are powers of 2 in all partitions of n.
%F A342231 G.f.: Sum_{k>=0} 2^k*x^(2^k)/(1 - x^(2^k)) / Product_{j>=1} (1 - x^j).
%F A342231 a(n) = Sum_{k=1..n} A038712(k) * A000041(n-k).
%e A342231 For n = 4 we have:
%e A342231 ------------------------------------
%e A342231 Partitions          Sum of parts
%e A342231 .              which are powers of 2
%e A342231 ------------------------------------
%e A342231 4 ..................... 4
%e A342231 3 + 1 ................. 1
%e A342231 2 + 2 ................. 4
%e A342231 2 + 1 + 1 ............. 4
%e A342231 1 + 1 + 1 + 1 ......... 4
%e A342231 ------------------------------------
%e A342231 Total ................ 17
%e A342231 So a(4) = 17.
%t A342231 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 A342231 Table[Sum[(2^IntegerExponent[2 k, 2] - 1) PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]
%Y A342231 Cf. A000041, A000079, A038712, A066186, A342230.
%K A342231 nonn
%O A342231 0,3
%A A342231 _Ilya Gutkovskiy_, Mar 06 2021