A363733 - OEIS

%I #22 Jun 29 2023 03:17:51

%S 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,

%T 93,68,30,6,1,0,2,78,246,276,130,42,7,1,0,4,130,768,1028,655,222,56,8,

%U 1,0,3,278,2190,4180,3130,1338,350,72,9,1

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

%C 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.

%C 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.

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

%F 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.

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

%e Array A(n, k) starts:

%e   [0] 1, 1,   1,    1,     1,      1,       1,       1,        1, ... A000012

%e   [1] 0, 1,   2,    3,     4,      5,       6,       7,        8, ... A001477

%e   [2] 0, 2,   6,   12,    20,     30,      42,      56,       72, ... A002378

%e   [3] 0, 2,  10,   30,    68,    130,     222,     350,      520, ... A034262

%e   [4] 0, 3,  22,   93,   276,    655,    1338,    2457,     4168, ...

%e   [5] 0, 2,  34,  246,  1028,   3130,    7782,   16814,    32776, ...

%e   [6] 0, 4,  78,  768,  4180,  15780,   46914,  118048,   262728, ...

%e   [7] 0, 2, 130, 2190, 16388,  78130,  279942,  823550,  2097160, ...

%e   [8] 0, 4, 278, 6654, 65812, 391280, 1680954, 5767258, 16781384, ...

%e    A000005,A055895,A363913, ...                             A066108 (diagonal)

%e .

%e Triangle T(n, k) starts:

%e   [0] 1;

%e   [1] 0, 1;

%e   [2] 0, 1,   1;

%e   [3] 0, 2,   2,   1;

%e   [4] 0, 2,   6,   3,    1;

%e   [5] 0, 3,  10,  12,    4,   1;

%e   [6] 0, 2,  22,  30,   20,   5,   1;

%e   [7] 0, 4,  34,  93,   68,  30,   6,  1;

%e   [8] 0, 2,  78, 246,  276, 130,  42,  7, 1;

%e   [9] 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1;

%p divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):

%p A := (n, k) -> local j; add(divides(j, n) * k^j, j = 0 ..n):

%p for n from 0 to 8 do seq(A(n, k), k = 0..8) od;

%p # If we introduce the 'inverse Möbius transform' InvMoebius acting on s ...

%p InvMoebius := (s, n) -> local j; add(divides(j, n) * s(j), j = 0 ..n):

%p # ... the transposed array is given by applying InvMoebius to the powers r^m:

%p seq(lprint(seq(InvMoebius(m -> r^m, n), n = 0..8)), r = 0..8);

%p # For instance we see that the number of divisors is the inverse

%p # Moebius transform of the constant sequence s = 1.

%o (SageMath)

%o def A(n, k): return sum(j.divides(n) * k^j for j in (0..n))

%o for n in srange(9): print([A(n, k) for k in (0..8)])

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

%Y Cf. A363734, A363735, A363421.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Jun 27 2023