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A342230 Total number of parts which are powers of 2 in all partitions of n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
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
Sequence in context: A078864 A208574 A361275 * A023218 A073022 A129694
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 06 2021
STATUS
approved

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Last modified February 29 02:52 EST 2024. Contains 370401 sequences. (Running on oeis4.)