login
A292193
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j).
8
1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 14, 7, 1, 1, 17, 36, 46, 25, 11, 1, 1, 33, 98, 164, 107, 56, 15, 1, 1, 65, 276, 610, 505, 352, 97, 22, 1, 1, 129, 794, 2324, 2531, 2474, 789, 198, 30, 1, 1, 257, 2316, 8986, 13225, 18580, 7273, 2314, 354, 42
OFFSET
0,6
LINKS
FORMULA
A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 02 2017
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, ...
2, 3, 5, 9, 17, ...
3, 6, 14, 36, 98, ...
5, 14, 46, 164, 610, ...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
`if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 11 2017
MATHEMATICA
m = 12;
col[k_] := col[k] = Product[1/(1 - j^k*x^j), {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)
CROSSREFS
Columns k=0..5 give A000041, A006906, A077335, A265837, A265838, A265839.
Rows 0+1, 2 give A000012, A000051.
Main diagonal gives A292194.
Cf. A292166.
Sequence in context: A050446 A214868 A144048 * A258708 A113983 A199333
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Sep 11 2017
STATUS
approved