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A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

%I A373301 #10 Jun 04 2024 14:24:25
%S A373301 0,3,12,40,98,253,540,1199,2415,4893,9268,17864,32421,59265,104632,
%T A373301 184338,315414,540155,901845,1504173,2461932,4013511,6443170,10314675,
%U A373301 16281749,25608450,39838855,61716941,94682665,144726102
%N A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.
%C A373301 The length of each row is given by A000041.
%C A373301 As many sequences start like the nonnegative integers, their row sums when disposed in this shape start with the same values.
%C A373301 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 A373301 Illustration of the first few terms
%e A373301 .
%e A373301 0   | 0
%e A373301 3   | 1,  2
%e A373301 12  | 3,  4,  5
%e A373301 40  | 6,  7,  8,  9,  10
%e A373301 98  | 11, 12, 13, 14, 15, 16, 17
%e A373301 253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
%e A373301 540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43
%e A373301 .
%t A373301 Module[{s = -1},
%t A373301  Table[s +=
%t A373301    PartitionsP[
%t A373301     n - 1]; (s + PartitionsP[n]) (s + PartitionsP[n] - 1)/2 -
%t A373301    s (s - 1)/2, {n, 1, 30}]]
%Y A373301 Cf. A373300, original version, with positive integers A000027.
%Y A373301 Cf. A001477, the nonnegative integers.
%Y A373301 Cf. A027480, the sequence of row sums for a regular triangle.
%K A373301 nonn
%O A373301 1,2
%A A373301 _Olivier Gérard_, May 31 2024