A342230 Total number of parts which are powers of 2 in all partitions of n.

0, 1, 3, 5, 11, 17, 29, 44, 71, 102, 153, 216, 311, 429, 599, 810, 1108, 1475, 1974, 2595, 3421, 4441, 5776, 7422, 9542, 12147, 15459, 19513, 24617, 30838, 38590, 48012, 59662, 73754, 91056, 111916, 137357, 167922, 204982, 249349, 302873, 366732, 443390, 534573

Offset: 0

Author

Ilya Gutkovskiy, Mar 06 2021

Keywords

• nonn

Examples

			For n = 4 we have:
------------------------------------
Partitions        Number of parts
.              which are powers of 2
------------------------------------
4 ..................... 1
3 + 1 ................. 1
2 + 2 ................. 2
2 + 1 + 1 ............. 3
1 + 1 + 1 + 1 ......... 4
------------------------------------
Total ................ 11
So a(4) = 11.
		

Crossrefs

Cf. A000041, A000070, A000079, A001511, A006128, A073119, A342231.

Programs

Mathematica

nmax = 43; CoefficientList[Series[Sum[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[IntegerExponent[2 k, 2] PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]

Formula

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

a(n) = Sum_{k=1..n} A001511(k) * A000041(n-k).

a(n) = A000070(n-1) + A073119(n).