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

%I #10 Jun 04 2024 14:24:25

%S 0,3,12,40,98,253,540,1199,2415,4893,9268,17864,32421,59265,104632,

%T 184338,315414,540155,901845,1504173,2461932,4013511,6443170,10314675,

%U 16281749,25608450,39838855,61716941,94682665,144726102

%N Sum of successive nonnegative 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 nonnegative 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 A001477 to have the same row sums for at least 8 terms: A089867, A089868, A089869, A089870, A118760, A123719, A130696, A136602, A254109, A258069, A258070, A258071, A266279, A272813, A273885, A273886, A273887, A273888.

%e Illustration of the first few terms

%e .

%e 0 | 0

%e 3 | 1, 2

%e 12 | 3, 4, 5

%e 40 | 6, 7, 8, 9, 10

%e 98 | 11, 12, 13, 14, 15, 16, 17

%e 253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

%e 540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43

%e .

%t Module[{s = -1},

%t Table[s +=

%t PartitionsP[

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

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

%Y Cf. A373300, original version, with positive integers A000027.

%Y Cf. A001477, the nonnegative integers.

%Y Cf. A027480, the sequence of row sums for a regular triangle.

%K nonn

%O 1,2

%A _Olivier GĂ©rard_, May 31 2024