A340493 Sequence whose partial sums give A340492.

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

Author

Omar E. Pol, Jan 10 2021

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Partial sums give A340492.

Cf. A000041, A000070, A090982, A336811.

Programs

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 *)

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

Formula

a(1) = 1.

a(n) = A000041(n)*A000070(n-1) - A000041(n-1)*A000070(n-2), n >= 2.