OFFSET
0,4
COMMENTS
Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*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) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|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, ...
2, 4, 11, 36, 135, 566, 2610, 13082, 70631, ...
3, 7, 21, 74, 296, 1315, 6393, 33645, 190085, ...
5, 12, 38, 141, 592, 2752, 13960, 76464, 448603, ...
7, 19, 64, 250, 1098, 5317, 28009, 158926, 963913, ...
11, 30, 105, 426, 1940, 9722, 52902, 309546, 1933171, ...
15, 45, 165, 696, 3281, 16972, 95129, 572402, 3670878, ...
22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
...
MAPLE
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,
combinat[numbpart](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..10);
MATHEMATICA
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, PartitionsP[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, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 16 2021
STATUS
approved