|
%I #34 Feb 03 2021 23:33:13
%S 1,3,8,23,49,125,241,540,1020,2064,3710,7231,12457,22883,39053,68596,
%T 113751,194865,315910,526019,840939,1363524,2144528,3419185,5291079,
%U 8277252,12668264,19497436,29459144,44762200,66847518,100267761,148318881,219818270,322056529,472600353
%N Sequence whose partial sums give A340492.
%C In other words: 1 together with the first differences of A340492.
%C Conjecture: a(n) is the size of the n-th section of a table of correspondence between divisors and partitions.
%F a(1) = 1.
%F a(n) = A000041(n)*A000070(n-1) - A000041(n-1)*A000070(n-2), n >= 2.
%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 region (or section) of the diagram.
%e For n = 3 the third region of the diagram contains 8 cells, so a(3) = 8.
%e For n = 7 the seventh region of the diagram contains 241 cells, so a(7) = 241.
%t 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 *)
%o (PARI) f(n) = numbpart(n)*sum(k=0, n-1, numbpart(k)); \\ A340492
%o a(n) = if (n==1, 1, f(n)-f(n-1)); \\ _Michel Marcus_, Jan 28 2021
%Y Partial sums give A340492.
%Y Cf. A000041, A000070, A090982, A336811.
%K nonn
%O 1,2
%A _Omar E. Pol_, Jan 10 2021
| 