This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365383 #6 Sep 09 2023 06:55:49
%S A365383 1,2,1,3,2,2,5,3,4,3,7,5,6,6,6,11,7,9,8,9,7,15,11,13,13,14,13,14,22,
%T A365383 15,19,17,20,17,20,16,30,22,26,26,27,26,28,26,27,42,30,37,34,39,33,40,
%U A365383 34,39,34,56,42,50,49,52,50,54,51,54,53,53
%N A365383 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be linearly combined with nonnegative coefficients to obtain k.
%C A365383 Conjecture: The rows eventually become periodic with period n if extended further. For example, row n = 8 begins:
%C A365383 22, 15, 19, 17, 20, 17, 20, 16,
%C A365383 22, 17, 20, 17, 21, 17, 20, 17,
%C A365383 22, 17, 20, 17, 21, 17, 20, 17, ...
%e A365383 Triangle begins:
%e A365383 1
%e A365383 2 1
%e A365383 3 2 2
%e A365383 5 3 4 3
%e A365383 7 5 6 6 6
%e A365383 11 7 9 8 9 7
%e A365383 15 11 13 13 14 13 14
%e A365383 22 15 19 17 20 17 20 16
%e A365383 30 22 26 26 27 26 28 26 27
%e A365383 42 30 37 34 39 33 40 34 39 34
%e A365383 56 42 50 49 52 50 54 51 54 53 53
%e A365383 77 56 68 64 71 63 73 63 71 65 70 62
%e A365383 101 77 91 89 95 90 97 93 97 97 98 94 99
%e A365383 135 101 122 115 127 115 130 114 131 119 130 117 132 116
%e A365383 176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
%e A365383 Row n = 6 counts the following partitions:
%e A365383 (6) (51) (51) (51) (51) (51)
%e A365383 (51) (411) (42) (411) (42) (411)
%e A365383 (42) (321) (411) (33) (411) (321)
%e A365383 (411) (3111) (321) (321) (321) (3111)
%e A365383 (33) (2211) (3111) (3111) (3111) (2211)
%e A365383 (321) (21111) (222) (2211) (222) (21111)
%e A365383 (3111) (111111) (2211) (21111) (2211) (111111)
%e A365383 (222) (21111) (111111) (21111)
%e A365383 (2211) (111111) (111111)
%e A365383 (21111)
%e A365383 (111111)
%t A365383 combu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365383 Table[Length[Select[IntegerPartitions[n],combu[k,#]!={}&]],{n,0,12},{k,0,n-1}]
%Y A365383 Column k = 0 is A000041, strict A000009.
%Y A365383 The version for subsets is A365381, main diagonal A365376.
%Y A365383 A000041 counts integer partitions, strict A000009.
%Y A365383 A008284 counts partitions by length, strict A008289.
%Y A365383 A116861 and A364916 count linear combinations of strict partitions.
%Y A365383 A364350 counts combination-free strict partitions, non-strict A364915.
%Y A365383 A364839 counts combination-full strict partitions, non-strict A364913.
%Y A365383 Cf. A088314, A088528, A237668, A363225, A364345, A365073, A365311, A365320, A365378, A365382.
%K A365383 nonn,tabl
%O A365383 0,2
%A A365383 _Gus Wiseman_, Sep 08 2023