%I #29 Feb 10 2021 03:41:16

%S 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,

%T 97,119,83,28,5,1,1,64,275,457,433,201,49,6,1,1,128,793,1763,2297,

%U 1476,487,83,8,1,1,256,2315,6841,12421,11113,4962,1141,142,10,1,1

%N 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.

%H Seiichi Manyama, <a href="/A284992/b284992.txt">Antidiagonals n = 0..139, flattened</a>

%F G.f. of column k: Product_{j>=1} (1+x^j)^(j^k).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 4, 8, 16, 32, 64, 128, ...

%e 2, 5, 13, 35, 97, 275, 793, 2315, ...

%e 2, 8, 31, 119, 457, 1763, 6841, 26699, ...

%e 3, 16, 83, 433, 2297, 12421, 68393, 382573, ...

%e 4, 28, 201, 1476, 11113, 85808, 678101, 5466916, ...

%e 5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))

%p end:

%p A:= (n, k)-> b(n$2, k):

%p seq(seq(A(n, d-n), n=0..d), d=0..14); # _Alois P. Heinz_, Oct 16 2017

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,

%t Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]];

%t A[n_, k_] := b[n, n, k];

%t Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 10 2021, after _Alois P. Heinz_ *)

%Y Columns k=0-5 give A000009, A026007, A027998, A248882, A248883, A248884.

%Y Rows (0+1),2-3 give: A000012, A000079, A007689.

%Y Main diagonal gives A270917.

%Y Cf. A283272, A284993.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Apr 07 2017