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A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

%I A373300 #11 Jun 04 2024 19:15:28
%S A373300 1,5,15,45,105,264,555,1221,2445,4935,9324,17941,32522,59400,104808,
%T A373300 184569,315711,540540,902335,1504800,2462724,4014513,6444425,10316250,
%U A373300 16283707,25610886,39841865,61720659,94687230,144731706,219282679,330996105,495901413,740046425
%N A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.
%C A373300 The length of each row is given by A000041.
%C A373300 As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
%C A373300 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 A373300 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 A373300 Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
%e A373300 .
%e A373300     1 |  1
%e A373300     5 |  2  3
%e A373300    15 |  4  5  6
%e A373300    45 |  7  8  9 10 11
%e A373300   105 | 12 13 14 15 16 17 18
%e A373300   264 | 19 20 21 22 23 24 25 26 27 28 29
%e A373300   555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
%e A373300 .
%e A373300 The sequence gives the row sums of this triangle.
%t A373300 Module[{s = 0},
%t A373300  Table[s +=
%t A373300    PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
%t A373300    s*(s - 1)/2, {n, 1, 30}]]
%Y A373300 Cf. A000027, seen as a triangle with shape A000041.
%Y A373300 Cf. A373301, the same principle, but starting from integer zero instead of 1.
%Y A373300 Cf. A006003, row sums of the integers but for the linear triangle.
%K A373300 nonn
%O A373300 1,2
%A A373300 _Olivier Gérard_, May 31 2024