%I #17 Jul 09 2024 19:41:38

%S 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 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 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 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 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 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 Alois P. Heinz, <a href="/A364916/b364916.txt">Antidiagonals n = 0..200, flattened</a>

%e Array begins:

%e 1 1 1 2 2 3 4 5 6 8 10 12 15 18 22 27

%e 0 1 0 1 1 1 2 2 3 3 5 5 7 8 10 12

%e 0 1 1 2 1 2 4 4 5 6 9 10 13 15 19 23

%e 0 1 0 3 2 2 4 4 6 7 11 11 15 17 22 27

%e 0 1 1 3 3 3 7 7 8 10 16 17 23 27 33 42

%e 0 1 0 3 2 4 7 6 9 9 17 17 23 26 33 43

%e 0 1 1 5 3 4 12 10 13 16 26 27 36 42 52 68

%e 0 1 0 4 3 3 10 11 13 13 27 25 35 40 51 67

%e 0 1 1 5 4 5 15 13 19 20 36 37 51 58 72 97

%e 0 1 0 6 4 5 14 13 18 23 42 39 54 61 78 105

%e 0 1 1 6 4 6 20 17 23 25 54 50 69 80 98 138

%e 0 1 0 6 4 5 19 16 23 24 54 55 71 80 103 144

%e 0 1 1 8 6 7 27 23 30 35 72 70 103 113 139 199

%e 0 1 0 7 5 6 24 21 29 31 75 68 95 115 139 201

%e 0 1 1 8 5 7 31 27 36 39 90 86 122 137 178 255

%e 0 1 0 9 6 8 31 27 38 42 100 93 129 148 187 289

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 2 0 1 0

%e 2 1 1 1 0

%e 3 1 2 0 1 0

%e 4 1 1 3 1 1 0

%e 5 2 2 2 3 0 1 0

%e 6 2 4 2 3 3 1 1 0

%e 8 3 4 4 3 2 5 0 1 0

%e 10 3 5 4 7 4 3 4 1 1 0

%e 12 5 6 6 7 7 4 3 5 0 1 0

%e 15 5 9 7 8 6 12 3 4 6 1 1 0

%e 18 7 10 11 10 9 10 10 5 4 6 0 1 0

%e 22 8 13 11 16 9 13 11 15 5 4 6 1 1 0

%e 27 10 15 15 17 17 16 13 13 14 6 4 8 0 1 0

%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];

%t t[n_,k_]:=Length[Join@@Table[combs[n,ptn],{ptn,Select[IntegerPartitions[k],UnsameQ@@#&]}]];

%t Table[t[k,n-k],{n,0,15},{k,0,n}]

%Y Same as A116861 with offset 0 and rows reversed, non-strict version A364912.

%Y Row n = 0 is A000009.

%Y Row n = 1 is A096765.

%Y Row n = 2 is A365005.

%Y Column k = 0 is A000007.

%Y Column k = 1 is A000012.

%Y Column k = 2 is A000035.

%Y Column k = 3 is A137719.

%Y The main diagonal is A364910.

%Y Left half has row sums A365002.

%Y For not just strict partitions we have A365004, diagonal A364907.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A066328 adds up distinct prime indices.

%Y A364350 counts combination-free strict partitions, complement A364839.

%Y Cf. A007865, A085489, A108917, A237113, A323092, A364272, A364533, A364670, A364911, A364913.

%K nonn,tabl

%O 0,7

%A _Gus Wiseman_, Aug 17 2023