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 A346500 #35 Aug 18 2021 15:28:28
%S A346500 1,1,1,2,2,2,5,4,4,5,15,11,9,11,15,52,36,26,26,36,52,203,135,92,66,92,
%T A346500 135,203,877,566,371,249,249,371,566,877,4140,2610,1663,1075,712,1075,
%U A346500 1663,2610,4140,21147,13082,8155,5133,3274,3274,5133,8155,13082,21147
%N A346500 Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A346500 Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i); A(2,2) = 9: 2*2*3*3, 3*3*4, 6*6, 2*3*6, 4*9, 2*2*9, 3*12, 2*18, 36.
%H A346500 Alois P. Heinz, <a href="/A346500/b346500.txt">Antidiagonals n = 0..140, flattened</a>
%F A346500 A(n,k) = A001055(A002110(n)*A002110(k)).
%F A346500 A(n,k) = A(k,n).
%F A346500 A(n,k) = A322765(abs(n-k),min(n,k)).
%e A346500 A(2,2) = 9: 1122, 11|22, 12|12, 1|122, 112|2, 11|2|2, 1|1|22, 1|12|2, 1|1|2|2.
%e A346500 Square array A(n,k) begins:
%e A346500 1, 1, 2, 5, 15, 52, 203, 877, ...
%e A346500 1, 2, 4, 11, 36, 135, 566, 2610, ...
%e A346500 2, 4, 9, 26, 92, 371, 1663, 8155, ...
%e A346500 5, 11, 26, 66, 249, 1075, 5133, 26683, ...
%e A346500 15, 36, 92, 249, 712, 3274, 16601, 91226, ...
%e A346500 52, 135, 371, 1075, 3274, 10457, 56135, 325269, ...
%e A346500 203, 566, 1663, 5133, 16601, 56135, 198091, 1207433, ...
%e A346500 877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, ...
%e A346500 ...
%p A346500 g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
%p A346500 `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
%p A346500 g(n/d, d)), d=divisors(n) minus {1, n}))
%p A346500 end:
%p A346500 p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
%p A346500 A:= (n, k)-> g(p(n)*p(k)$2):
%p A346500 seq(seq(A(n, d-n), n=0..d), d=0..10);
%p A346500 # second Maple program:
%p A346500 b:= proc(n) option remember; `if`(n=0, 1,
%p A346500 add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A346500 end:
%p A346500 A:= proc(n, k) option remember; `if`(n<k, A(k, n),
%p A346500 `if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)
%p A346500 *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
%p A346500 end:
%p A346500 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A346500 b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
%t A346500 A[n_, k_] := A[n, k] = If[n < k, A[k, n],
%t A346500 If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]*
%t A346500 Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
%t A346500 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Aug 18 2021, after _Alois P. Heinz_'s second program *)
%Y A346500 Columns (or rows) k=0-10 give: A000110, A035098, A322764, A322768, A346881, A346882, A346883, A346884, A346885, A346886, A346887.
%Y A346500 Main diagonal gives A020555.
%Y A346500 First upper (or lower) diagonal gives A322766.
%Y A346500 Second upper (or lower) diagonal gives A322767.
%Y A346500 Antidiagonal sums give A346490.
%Y A346500 A(2n,n) gives A322769.
%Y A346500 Cf. A001055, A002110, A322765, A346426, A346517.
%K A346500 nonn,tabl
%O A346500 0,4
%A A346500 _Alois P. Heinz_, Jul 20 2021