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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

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-Fran├žois 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.

Cf. A283272, A284993.

Sequence in context: A229054 A133135 A292189 * A191687 A322190 A177254

Adjacent sequences:  A284989 A284990 A284991 * A284993 A284994 A284995

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Apr 07 2017

STATUS

approved

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)