A382881 Triangle read by rows: T(n, k) = -Sum_{d|n, d<n} V(n, d)*T(d, k) for k >= 1, T(n, 0) = 0^n, T(n, n) = 1, where V(n, d) = 1 if d = 1 otherwise valuation(n, d).

1, 0, 1, 0, -1, 1, 0, -1, 0, 1, 0, 1, -2, 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, -1, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, -2, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1

Offset

0,13

Comments

See the comments in A382883.

Links

Table of n, a(n) for n=0..90.

Formula

T is the matrix inverse of A382944.

Example

Triangle start:
  [0] 1;
  [1] 0,  1;
  [2] 0, -1,  1;
  [3] 0, -1,  0,  1;
  [4] 0,  1, -2,  0,  1;
  [5] 0, -1,  0,  0,  0, 1;
  [6] 0,  1, -1, -1,  0, 0, 1;
  [7] 0, -1,  0,  0,  0, 0, 0, 1;
  [8] 0,  1, -1,  0, -1, 0, 0, 0, 1;
  [9] 0,  1,  0, -2,  0, 0, 0, 0, 0, 1;

Maple

A382881 := proc(n, k) option remember; if n = k then 1 elif k = 0 then 0^n else -add(ifelse(j = 1, 1, padic:-ordp(n, j))*A382881(j, k), j in NumberTheory:-Divisors(n) minus {n}) fi end: seq(seq(A382881(n, k), k = 0..n), n = 0..9);

Mathematica

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; T[n_, 0] := T[n, 0] = Boole[n == 0]; T[n_, n_] := T[n, n] = 1; T[n_, k_] := T[n, k] = -DivisorSum[n, V[n, #] * T[#, k] &, # < n &]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2025 *)

Prog

(SageMath)
@cached_function
def T(n, k):
    if n == k: return 1
    if k == 0: return 0^n
    return -sum((1 if j == 1 else valuation(n, j))*T(j, k) for j in divisors(n)[:-1])
for n in range(20): print([T(n, k) for k in srange(n + 1)])

Crossrefs

Cf. A382944, A382883 (column 1), A019590 (with offset 0 the row sums).

Sequence in context: A372505 A322338 A158971 * A382944 A121467 A386262

Adjacent sequences: A382878 A382879 A382880 * A382882 A382883 A382884

Keyword

sign,tabl

Author

Peter Luschny, Apr 09 2025

Status

approved