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A340493
Sequence whose partial sums give A340492.
1
1, 3, 8, 23, 49, 125, 241, 540, 1020, 2064, 3710, 7231, 12457, 22883, 39053, 68596, 113751, 194865, 315910, 526019, 840939, 1363524, 2144528, 3419185, 5291079, 8277252, 12668264, 19497436, 29459144, 44762200, 66847518, 100267761, 148318881, 219818270, 322056529, 472600353
OFFSET
1,2
COMMENTS
In other words: 1 together with the first differences of A340492.
Conjecture: a(n) is the size of the n-th section of a table of correspondence between divisors and partitions.
FORMULA
a(1) = 1.
a(n) = A000041(n)*A000070(n-1) - A000041(n-1)*A000070(n-2), n >= 2.
EXAMPLE
Illustration of initial terms:
A000070: 1 2 4 7 12 19 30
A000041 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1 |_| | | | | | |
2 |_ _| | | | | |
3 |_ _ _ _| | | | |
| | | | |
5 |_ _ _ _ _ _ _| | | |
| | | |
7 |_ _ _ _ _ _ _ _ _ _ _ _| | |
| | |
| | |
| | |
11 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
| |
| |
| |
15 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
...
a(n) is the area (or the number of cells) in the n-th region (or section) of the diagram.
For n = 3 the third region of the diagram contains 8 cells, so a(3) = 8.
For n = 7 the seventh region of the diagram contains 241 cells, so a(7) = 241.
MATHEMATICA
a[n_] := PartitionsP[n]*Count[Flatten[IntegerPartitions[n]], 1] - PartitionsP[n - 1]*Count[Flatten[IntegerPartitions[n - 1]], 1]; Table[a[n], {n, 1, 36}] (* Robert P. P. McKone, Jan 28 2021 *)
PROG
(PARI) f(n) = numbpart(n)*sum(k=0, n-1, numbpart(k)); \\ A340492
a(n) = if (n==1, 1, f(n)-f(n-1)); \\ Michel Marcus, Jan 28 2021
CROSSREFS
Partial sums give A340492.
Sequence in context: A073051 A183930 A183922 * A296286 A068602 A255834
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 10 2021
STATUS
approved