A383033 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).

1, 1, 1, 1, 2, 0, 1, 3, 2, 1, 1, 4, 6, 5, 0, 1, 5, 12, 18, 6, 0, 1, 6, 20, 46, 42, 11, 0, 1, 7, 30, 95, 156, 113, 18, 1, 1, 8, 42, 171, 420, 566, 294, 35, 0, 1, 9, 56, 280, 930, 1930, 2028, 798, 56, 0, 1, 10, 72, 428, 1806, 5185, 8820, 7396, 2128, 105, 0

Offset

1,5

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

A(n,k) = A383023(n,k) - A383023(n,k-1).

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

Example

Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  1,  2,   3,    4,    5,     6,     7, ...
  0,  2,   6,   12,   20,    30,    42, ...
  1,  5,  18,   46,   95,   171,   280, ...
  0,  6,  42,  156,  420,   930,  1806, ...
  0, 11, 113,  566, 1930,  5185, 11851, ...
  0, 18, 294, 2028, 8820, 28830, 77658, ...

Crossrefs

Columns k=1..3 give A209229, A383034, A383035.

Main diagonal gives A316073.

Cf. A383023.

Sequence in context: A263390 A231354 A197119 * A124377 A144190 A239689

Adjacent sequences: A383030 A383031 A383032 * A383034 A383035 A383036

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 13 2025

Status

approved