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A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.

%I A365004 #15 Jan 28 2024 20:41:21
%S A365004 1,1,0,2,1,0,3,2,1,0,5,4,4,1,0,7,7,8,4,1,0,11,12,17,13,6,1,0,15,19,30,
%T A365004 28,18,6,1,0,22,30,53,58,50,24,8,1,0,30,45,86,109,108,70,33,8,1,0,42,
%U A365004 67,139,194,223,179,107,40,10,1,0,56,97,213,328,420,394,286,143,50,10,1,0
%N A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.
%C A365004 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).
%H A365004 Alois P. Heinz, <a href="/A365004/b365004.txt">Rows n = 0..200, flattened</a>
%F A365004 Also the number of ways to write n-k as a *positive* linear combination of an integer partition of k.
%e A365004 Array begins:
%e A365004   1  1  2   3   5    7     11
%e A365004   0  1  2   4   7    12    19
%e A365004   0  1  4   8   17   30    53
%e A365004   0  1  4   13  28   58    109
%e A365004   0  1  6   18  50   108   223
%e A365004   0  1  6   24  70   179   394
%e A365004   0  1  8   33  107  286   696
%e A365004   0  1  8   40  143  428   1108
%e A365004   0  1  10  50  199  628   1754
%e A365004   0  1  10  61  254  882   2622
%e A365004   0  1  12  72  332  1215  3857
%e A365004   0  1  12  84  410  1624  5457
%e A365004   0  1  14  99  517  2142  7637
%e A365004 The A(4,2) = 6 ways:
%e A365004   2*2
%e A365004   0*1+4*1
%e A365004   1*1+3*1
%e A365004   2*1+2*1
%e A365004   3*1+1*1
%e A365004   4*1+0*1
%p A365004 b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
%p A365004      `if`(i<1, 0, b(n, i-1, m)+add(b(n-i, min(i, n-i), m-i*j), j=0..m/i)))
%p A365004     end:
%p A365004 A:= (n, k)-> b(k$2, n):
%p A365004 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Jan 28 2024
%t A365004 nn=5;
%t A365004 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365004 tabv=Table[Length[Join@@Table[combs[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,nn},{k,0,nn}]
%t A365004 Table[tabv[[k+1,n-k+1]],{n,0,nn},{k,0,n}]
%Y A365004 Row n = 0 is A000041, strict A000009.
%Y A365004 Row n = 1 is A000070.
%Y A365004 Column k = 0 is A000007.
%Y A365004 Column k = 1 is A000012.
%Y A365004 Column k = 2 is A052928 except initial terms.
%Y A365004 Antidiagonal sums are A006951.
%Y A365004 The case of strict integer partitions is A116861.
%Y A365004 Main diagonal is A364907.
%Y A365004 The transpose is A364912, also the positive version.
%Y A365004 A008284 counts partitions by length, strict A008289.
%Y A365004 A364350 counts combination-free strict partitions, complement A364839.
%Y A365004 A364913 counts combination-full partitions.
%Y A365004 Cf. A066328, A108917, A237113, A364272, A364910, A364911, A364915, A365002.
%K A365004 nonn,tabl
%O A365004 0,4
%A A365004 _Gus Wiseman_, Aug 23 2023
%E A365004 Antidiagonals 8-11 from _Alois P. Heinz_, Jan 28 2024