A321163 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the sum of distinct products Product_{j=1..k} b_j with 1 <= b_j<= n.

1, 1, 3, 1, 7, 6, 1, 15, 25, 10, 1, 31, 90, 61, 15, 1, 63, 301, 310, 136, 21, 1, 127, 966, 1441, 990, 244, 28, 1, 255, 3025, 6370, 6391, 2220, 440, 36, 1, 511, 9330, 27301, 38325, 17731, 5300, 680, 45, 1, 1023, 28501, 114670, 218926, 130851, 54831, 9660, 1022, 55

Offset

1,3

Links

Seiichi Manyama, Antidiagonals n = 1..25, flattened

Example

In case of (n,k) = (3,2):
  | 1  2  3
--+--------
1 | 1, 2, 3
2 | 2, 4, 6
3 | 3, 6, 9
Distinct products are 1,2,3,4,6,9. So A(3,2) = 1+2+3+4+6+9 = 25.
Square array begins:
    1,    1,     1,      1,       1,        1, ...
    3,    7,    15,     31,      63,      127, ...
    6,   25,    90,    301,     966,     3025, ...
   10,   61,   310,   1441,    6370,    27301, ...
   15,  136,   990,   6391,   38325,   218926, ...
   21,  244,  2220,  17731,  130851,   916714, ...
   28,  440,  5300,  54831,  514668,  4519390, ...
   36,  680,  9660, 116991, 1280916, 13092430, ...
   45, 1022, 17130, 242091, 3070935, 36184072, ...
   55, 1472, 28670, 467391, 6807045, 91765822, ...

Mathematica

A[n_, k_] := Module[{b, bb}, bb = Array[b, k]; Table[Times @@ bb, Evaluate[Sequence @@ ({#, n}& /@ bb)]] // Flatten // Union // Total];
Table[A[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 25 2020 *)

Crossrefs

Columns 1-3 give A000217, A321165, A323334.

Rows 1-2 give A000012, A000225(n+1).

Main diagonal gives A321164.

Cf. A322967.

Row 3 gives A000392(n+4). - Fred Daniel Kline, Jan 11 2019

Sequence in context: A110441 A111806 A372118 * A054458 A110168 A323663

Adjacent sequences: A321160 A321161 A321162 * A321164 A321165 A321166

Keyword

nonn,tabl

Author

Seiichi Manyama, Jan 10 2019

Status

approved