OFFSET
0,4
COMMENTS
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.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
EXAMPLE
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.
Square array A(n,k) begins:
1, 1, 2, 5, 15, 52, 203, 877, ...
1, 2, 4, 11, 36, 135, 566, 2610, ...
2, 4, 9, 26, 92, 371, 1663, 8155, ...
5, 11, 26, 66, 249, 1075, 5133, 26683, ...
15, 36, 92, 249, 712, 3274, 16601, 91226, ...
52, 135, 371, 1075, 3274, 10457, 56135, 325269, ...
203, 566, 1663, 5133, 16601, 56135, 198091, 1207433, ...
877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, ...
...
MAPLE
g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
`if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
g(n/d, d)), d=divisors(n) minus {1, n}))
end:
p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
A:= (n, k)-> g(p(n)*p(k)$2):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*binomial(n-1, j-1), j=1..n))
end:
A:= proc(n, k) option remember; `if`(n<k, A(k, n),
`if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)
*binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
A[n_, k_] := A[n, k] = If[n < k, A[k, n],
If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]*
Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
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 *)
CROSSREFS
Columns (or rows) k=0-10 give: A000110, A035098, A322764, A322768, A346881, A346882, A346883, A346884, A346885, A346886, A346887.
Main diagonal gives A020555.
First upper (or lower) diagonal gives A322766.
Second upper (or lower) diagonal gives A322767.
Antidiagonal sums give A346490.
A(2n,n) gives A322769.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 20 2021
STATUS
approved