A383023 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Weigh transform of j-> k^j.

1, 2, 1, 3, 3, 0, 4, 6, 2, 1, 5, 10, 8, 6, 0, 6, 15, 20, 24, 6, 0, 7, 21, 40, 70, 48, 11, 0, 8, 28, 70, 165, 204, 124, 18, 1, 9, 36, 112, 336, 624, 690, 312, 36, 0, 10, 45, 168, 616, 1554, 2620, 2340, 834, 56, 0, 11, 55, 240, 1044, 3360, 7805, 11160, 8230, 2184, 105, 0

Offset

1,2

Links

Table of n, a(n) for n=1..66.

Christian G. Bower, PARI programs for transforms, 2007.

N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Formula

A(n,k) = (1/n) * (k^n + Sum_{d<n and d|n} (-1)^(n/d) * d * A(d,k)).

Product_{n>=1} (1 + x^n)^A(n,k) = 1/(1 - k*x).

Example

Square array begins:
  1,  2,   3,    4,     5,     6,      7, ...
  1,  3,   6,   10,    15,    21,     28, ...
  0,  2,   8,   20,    40,    70,    112, ...
  1,  6,  24,   70,   165,   336,    616, ...
  0,  6,  48,  204,   624,  1554,   3360, ...
  0, 11, 124,  690,  2620,  7805,  19656, ...
  0, 18, 312, 2340, 11160, 39990, 117648, ...

Crossrefs

Columns k=1..5 give A209229, A306156, A306157, A306158, A306159.

Cf. A074650.

Sequence in context: A125210 A098434 A383011 * A375479 A212634 A162883

Adjacent sequences: A383020 A383021 A383022 * A383024 A383025 A383026

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 12 2025

Status

approved