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

0, 1, 4, 6, 17, 24, 43, 64, 115, 159, 247, 347, 513, 704, 1001, 1350, 1894, 2513, 3408, 4489, 5989, 7786, 10226, 13172, 17079, 21800, 27938, 35362, 44900, 56402, 70959, 88545, 110617, 137108, 170051, 209599, 258328, 316685, 388072, 473331, 577026, 700524, 849775, 1027167

Offset: 0

Author

Ilya Gutkovskiy, Mar 06 2021

Keywords

• nonn

Examples

			For n = 4 we have:
------------------------------------
Partitions          Sum of parts
.              which are powers of 2
------------------------------------
4 ..................... 4
3 + 1 ................. 1
2 + 2 ................. 4
2 + 1 + 1 ............. 4
1 + 1 + 1 + 1 ......... 4
------------------------------------
Total ................ 17
So a(4) = 17.
		

Crossrefs

Cf. A000041, A000079, A038712, A066186, A342230.

Programs

Mathematica

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]
Table[Sum[(2^IntegerExponent[2 k, 2] - 1) PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]

Formula

G.f.: Sum_{k>=0} 2^k*x^(2^k)/(1 - x^(2^k)) / Product_{j>=1} (1 - x^j).

a(n) = Sum_{k=1..n} A038712(k) * A000041(n-k).