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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.
8
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
0,9
LINKS
FORMULA
G.f. of column k: Product_{j>=1} (1+x^j)^(j^k).
EXAMPLE
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, ...
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 *)
CROSSREFS
Columns k=0-5 give A000009, A026007, A027998, A248882, A248883, A248884.
Rows (0+1),2-3 give: A000012, A000079, A007689.
Main diagonal gives A270917.
Sequence in context: A229054 A133135 A292189 * A191687 A322190 A177254
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Apr 07 2017
STATUS
approved