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 A364916 #17 Jul 09 2024 19:41:38
%S A364916 1,1,0,1,1,0,2,0,1,0,2,1,1,1,0,3,1,2,0,1,0,4,1,1,3,1,1,0,5,2,2,2,3,0,
%T A364916 1,0,6,2,4,2,3,3,1,1,0,8,3,4,4,3,2,5,0,1,0,10,3,5,4,7,4,3,4,1,1,0,12,
%U A364916 5,6,6,7,7,4,3,5,0,1,0,15,5,9,7,8,6,12,3,4,6,1,1,0
%N A364916 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.
%C A364916 A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).
%C A364916 As a triangle, also the number of ways to write n as a *positive* linear combination of the parts of a strict integer partition of k.
%H A364916 Alois P. Heinz, <a href="/A364916/b364916.txt">Antidiagonals n = 0..200, flattened</a>
%e A364916 Array begins:
%e A364916 1 1 1 2 2 3 4 5 6 8 10 12 15 18 22 27
%e A364916 0 1 0 1 1 1 2 2 3 3 5 5 7 8 10 12
%e A364916 0 1 1 2 1 2 4 4 5 6 9 10 13 15 19 23
%e A364916 0 1 0 3 2 2 4 4 6 7 11 11 15 17 22 27
%e A364916 0 1 1 3 3 3 7 7 8 10 16 17 23 27 33 42
%e A364916 0 1 0 3 2 4 7 6 9 9 17 17 23 26 33 43
%e A364916 0 1 1 5 3 4 12 10 13 16 26 27 36 42 52 68
%e A364916 0 1 0 4 3 3 10 11 13 13 27 25 35 40 51 67
%e A364916 0 1 1 5 4 5 15 13 19 20 36 37 51 58 72 97
%e A364916 0 1 0 6 4 5 14 13 18 23 42 39 54 61 78 105
%e A364916 0 1 1 6 4 6 20 17 23 25 54 50 69 80 98 138
%e A364916 0 1 0 6 4 5 19 16 23 24 54 55 71 80 103 144
%e A364916 0 1 1 8 6 7 27 23 30 35 72 70 103 113 139 199
%e A364916 0 1 0 7 5 6 24 21 29 31 75 68 95 115 139 201
%e A364916 0 1 1 8 5 7 31 27 36 39 90 86 122 137 178 255
%e A364916 0 1 0 9 6 8 31 27 38 42 100 93 129 148 187 289
%e A364916 Triangle begins:
%e A364916 1
%e A364916 1 0
%e A364916 1 1 0
%e A364916 2 0 1 0
%e A364916 2 1 1 1 0
%e A364916 3 1 2 0 1 0
%e A364916 4 1 1 3 1 1 0
%e A364916 5 2 2 2 3 0 1 0
%e A364916 6 2 4 2 3 3 1 1 0
%e A364916 8 3 4 4 3 2 5 0 1 0
%e A364916 10 3 5 4 7 4 3 4 1 1 0
%e A364916 12 5 6 6 7 7 4 3 5 0 1 0
%e A364916 15 5 9 7 8 6 12 3 4 6 1 1 0
%e A364916 18 7 10 11 10 9 10 10 5 4 6 0 1 0
%e A364916 22 8 13 11 16 9 13 11 15 5 4 6 1 1 0
%e A364916 27 10 15 15 17 17 16 13 13 14 6 4 8 0 1 0
%t A364916 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A364916 t[n_,k_]:=Length[Join@@Table[combs[n,ptn],{ptn,Select[IntegerPartitions[k],UnsameQ@@#&]}]];
%t A364916 Table[t[k,n-k],{n,0,15},{k,0,n}]
%Y A364916 Same as A116861 with offset 0 and rows reversed, non-strict version A364912.
%Y A364916 Row n = 0 is A000009.
%Y A364916 Row n = 1 is A096765.
%Y A364916 Row n = 2 is A365005.
%Y A364916 Column k = 0 is A000007.
%Y A364916 Column k = 1 is A000012.
%Y A364916 Column k = 2 is A000035.
%Y A364916 Column k = 3 is A137719.
%Y A364916 The main diagonal is A364910.
%Y A364916 Left half has row sums A365002.
%Y A364916 For not just strict partitions we have A365004, diagonal A364907.
%Y A364916 A000041 counts integer partitions, strict A000009.
%Y A364916 A008284 counts partitions by length, strict A008289.
%Y A364916 A066328 adds up distinct prime indices.
%Y A364916 A364350 counts combination-free strict partitions, complement A364839.
%Y A364916 Cf. A007865, A085489, A108917, A237113, A323092, A364272, A364533, A364670, A364911, A364913.
%K A364916 nonn,tabl
%O A364916 0,7
%A A364916 _Gus Wiseman_, Aug 17 2023