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

0,3

Links

Table of n, a(n) for 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).

Example

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.

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}]

Crossrefs

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

Sequence in context: A078864 A208574 A361275 * A023218 A073022 A129694

Adjacent sequences: A342227 A342228 A342229 * A342231 A342232 A342233

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 06 2021

Status

approved