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Sum of successive integers in a row of length p(n) where p counts integer partitions.
1

%I #11 Jun 04 2024 19:15:28

%S 1,5,15,45,105,264,555,1221,2445,4935,9324,17941,32522,59400,104808,

%T 184569,315711,540540,902335,1504800,2462724,4014513,6444425,10316250,

%U 16283707,25610886,39841865,61720659,94687230,144731706,219282679,330996105,495901413,740046425

%N Sum of successive integers in a row of length p(n) where p counts integer partitions.

%C The length of each row is given by A000041.

%C As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.

%C Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.

%C A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

%e Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.

%e .

%e 1 | 1

%e 5 | 2 3

%e 15 | 4 5 6

%e 45 | 7 8 9 10 11

%e 105 | 12 13 14 15 16 17 18

%e 264 | 19 20 21 22 23 24 25 26 27 28 29

%e 555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

%e .

%e The sequence gives the row sums of this triangle.

%t Module[{s = 0},

%t Table[s +=

%t PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -

%t s*(s - 1)/2, {n, 1, 30}]]

%Y Cf. A000027, seen as a triangle with shape A000041.

%Y Cf. A373301, the same principle, but starting from integer zero instead of 1.

%Y Cf. A006003, row sums of the integers but for the linear triangle.

%K nonn

%O 1,2

%A _Olivier GĂ©rard_, May 31 2024