A383042 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 Euler transform of j-> k^(j-1).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 15, 6, 0, 1, 5, 20, 42, 42, 9, 0, 1, 6, 30, 90, 156, 107, 18, 0, 1, 7, 42, 165, 420, 554, 294, 30, 0, 1, 8, 56, 273, 930, 1910, 2028, 780, 56, 0, 1, 9, 72, 420, 1806, 5155, 8820, 7350, 2128, 99, 0

Offset: 1

Author

Seiichi Manyama, Apr 13 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  2,   6,   12,   20,    30,    42, ...
  0,  3,  15,   42,   90,   165,   273, ...
  0,  6,  42,  156,  420,   930,  1806, ...
  0,  9, 107,  554, 1910,  5155, 11809, ...
  0, 18, 294, 2028, 8820, 28830, 77658, ...
  ...
		

Links

• Christian G. Bower, PARI programs for transforms, 2007.
• N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Crossrefs

Columns k=1..5 give A000007, A059966, A065178, A065179, A065180.

Main diagonal gives A306173.

Cf. A065177 (another version).

Cf. A008683, A074650, A383033.

Programs

PARI

a(n, k) = sumdiv(n, d, moebius(n/d)*(k^d-(k-1)^d))/n;

Formula

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

A(n,k) = (1/n) * (k^n - (k-1)^n - Sum_{d

A(n,k) = A074650(n,k) - A074650(n,k-1).

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

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + x^j/(1 - k*x^j)) / j.

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