A284992 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j^k) in powers of x.
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 8, 3, 1, 1, 16, 35, 31, 16, 4, 1, 1, 32, 97, 119, 83, 28, 5, 1, 1, 64, 275, 457, 433, 201, 49, 6, 1, 1, 128, 793, 1763, 2297, 1476, 487, 83, 8, 1, 1, 256, 2315, 6841, 12421, 11113, 4962, 1141, 142, 10, 1, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 4, 8, 16, 32, 64, 128, ... 2, 5, 13, 35, 97, 275, 793, 2315, ... 2, 8, 31, 119, 457, 1763, 6841, 26699, ... 3, 16, 83, 433, 2297, 12421, 68393, 382573, ... 4, 28, 201, 1476, 11113, 85808, 678101, 5466916, ... 5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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Maple
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i))) end: A:= (n, k)-> b(n$2, k): seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Oct 16 2017 -
Mathematica
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]]; A[n_, k_] := b[n, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)
Formula
G.f. of column k: Product_{j>=1} (1+x^j)^(j^k).