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

0,3

Links

Table of n, a(n) for 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).

Example

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.

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}]

Crossrefs

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

Sequence in context: A105271 A343758 A024305 * A320245 A337208 A034492

Adjacent sequences: A342228 A342229 A342230 * A342232 A342233 A342234

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 06 2021

Status

approved