OFFSET
0,4
COMMENTS
Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1) into distinct factors; A(3,1) = 5: 2*3*4, 4*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
EXAMPLE
A(2,2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|12, 0|012, 0012.
Square array A(n,k) begins:
1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
1, 3, 9, 31, 120, 514, 2407, 12205, 66491, ...
2, 5, 16, 59, 244, 1112, 5516, 29505, 168938, ...
2, 7, 25, 100, 442, 2134, 11147, 62505, 373832, ...
3, 10, 38, 161, 750, 3799, 20739, 121141, 752681, ...
4, 14, 56, 249, 1213, 6404, 36332, 220000, 1413937, ...
5, 19, 80, 372, 1887, 10340, 60727, 379831, 2516880, ...
6, 25, 111, 539, 2840, 16108, 97666, 629346, 4288933, ...
...
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
s:= proc(n) option remember; expand(`if`(n=0, 1,
x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))
end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, g[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j]*b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jul 31 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 21 2021
STATUS
approved