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%I #54 Feb 03 2021 23:32:47
%S 1,4,12,35,84,209,450,990,2010,4074,7784,15015,27472,50355,89408,
%T 158004,271755,466620,782530,1308549,2149488,3513012,5657540,9076725,
%U 14367804,22645056,35313320,54810756,84269900,129032100,195879618,296147379,444466260,664284530,986341059,1458941412
%N a(n) = A000041(n)*A000070(n-1), n >= 1.
%C Conjecture: a(n) is the size of a table of correspondence between the divisors of certain numbers and the partitions of n. The table contains A006128(n) positive integers and the rest are zeros (or empty cells). These positive integers can be interpreted in two ways as follows:
%C 1) They are all divisors of all terms of the first n rows of triangle A336811. In other words: they are all divisors of the first A000070(n-1) terms of A336811, hence they are all divisors of all terms in the n-th row of A176206.
%C 2) They are all parts of all partitions of n.
%C The mentioned divisors are in the columns of the table.
%C The mentioned partitions are in the rows of the table.
%C For more information about the main conjecture see A336811.
%e Illustration of initial terms:
%e A000070: 1 2 4 7 12 19 30
%e A000041 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 1 |_| | | | | | |
%e 2 |_ _| | | | | |
%e 3 |_ _ _ _| | | | |
%e | | | | |
%e 5 |_ _ _ _ _ _ _| | | |
%e | | | |
%e 7 |_ _ _ _ _ _ _ _ _ _ _ _| | |
%e | | |
%e | | |
%e | | |
%e 11 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e | |
%e | |
%e | |
%e 15 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e ...
%e a(n) is the area (or the number of cells) in the n-th rectangle of the diagram.
%e For n = 3 the third rectangle of the diagram contains 3*4 = 12 cells, so a(3) = 12
%e For n = 7 the seventh rectangle of the diagram contains 15*30 = 450 cells, so a(7) = 450.
%t a[n_] := PartitionsP[n]*Count[Flatten[IntegerPartitions[n]], 1]; Table[a[n], {n, 1, 36}] (* _Robert P. P. McKone_, Jan 28 2021 *)
%o (PARI) a(n) = numbpart(n)*sum(k=0, n-1, numbpart(k)); \\ _Michel Marcus_, Jan 28 2021
%Y Partial sums of A340493.
%Y Cf. A000041, A000070, A006128, A176206, A336811, A340061.
%K nonn
%O 1,2
%A _Omar E. Pol_, Jan 10 2021
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