A363733 Array read by upwards antidiagonals. The family of polynomials generated by the divisibility matrix (A113704) evaluated over the nonnegative integers.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 2, 22, 30, 20, 5, 1, 0, 4, 34, 93, 68, 30, 6, 1, 0, 2, 78, 246, 276, 130, 42, 7, 1, 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1, 0, 3, 278, 2190, 4180, 3130, 1338, 350, 72, 9, 1

Offset: 0

Author

Peter Luschny, Jun 27 2023

Keywords

• nonn
• tabl

Comments

The name expresses the 'row view' of the array. The 'column view' regards the array as the collection of the inverse Möbius transforms of the power sequences k^n = 0^n, 1^n, 2^n, .... (n >= 0). Viewed this way, the array is a generalization of the number of divisors sequence tau (A000005), to which it reduces in the case k = 1.

The array has offset (0, 0). It uses the usual definition of 'k divides n' as described in Apostol, rather than the shortened version, which restricts to values k > 0 as some programs do (but not SageMath). Such a restriction makes sense in the context of rational numbers but not in the case of natural numbers.

Examples

			Array A(n, k) starts:
  [0] 1, 1,   1,    1,     1,      1,       1,       1,        1, ... A000012
  [1] 0, 1,   2,    3,     4,      5,       6,       7,        8, ... A001477
  [2] 0, 2,   6,   12,    20,     30,      42,      56,       72, ... A002378
  [3] 0, 2,  10,   30,    68,    130,     222,     350,      520, ... A034262
  [4] 0, 3,  22,   93,   276,    655,    1338,    2457,     4168, ...
  [5] 0, 2,  34,  246,  1028,   3130,    7782,   16814,    32776, ... A131471
  [6] 0, 4,  78,  768,  4180,  15780,   46914,  118048,   262728, ...
  [7] 0, 2, 130, 2190, 16388,  78130,  279942,  823550,  2097160, ... A190578
  [8] 0, 4, 278, 6654, 65812, 391280, 1680954, 5767258, 16781384, ...
   A000005,A055895,A363913, ...                             A066108 (diagonal)
.
Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,   1;
  [3] 0, 2,   2,   1;
  [4] 0, 2,   6,   3,    1;
  [5] 0, 3,  10,  12,    4,   1;
  [6] 0, 2,  22,  30,   20,   5,   1;
  [7] 0, 4,  34,  93,   68,  30,   6,  1;
  [8] 0, 2,  78, 246,  276, 130,  42,  7, 1;
  [9] 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1;
		

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.

Crossrefs

Cf. A113704 (in compact form A113705), A000005 (column 1), A055895 (column 2), A363913 (column 3), A001477 (row 1), A002378 (row 2), A034262 (row 3), A131471 (row 5), A190578 (row 7), A363912 (row sums), A066108 (main diagonal of array).

Cf. A363734, A363735, A363421.

Programs

Maple

divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
A := (n, k) -> local j; add(divides(j, n) * k^j, j = 0 ..n):
for n from 0 to 8 do seq(A(n, k), k = 0..8) od;
# If we introduce the 'inverse Möbius transform' InvMoebius acting on s ...
InvMoebius := (s, n) -> local j; add(divides(j, n) * s(j), j = 0 ..n):
# ... the transposed array is given by applying InvMoebius to the powers r^m:
seq(lprint(seq(InvMoebius(m -> r^m, n), n = 0..8)), r = 0..8);
# For instance we see that the number of divisors is the inverse
# Moebius transform of the constant sequence s = 1.

SageMath

def A(n, k): return sum(j.divides(n) * k^j for j in (0..n))
for n in srange(9): print([A(n, k) for k in (0..8)])

Formula

A(n, k) = Sum_{j=0..n} divides(j, n) * k^j, where divides(k, n) <-> [k = n or (k > 0 and n mod k = 0)], and '[ ]' denotes the Iverson bracket.

The columns are the inverse Möbius transforms of the powers x^n, x >= 0.