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A363914
The Moebius triangle read by rows. Inverse matrix of the divisibility triangle A113704.
6
1, 0, 1, 0, -1, 1, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, -1, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 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
LINKS
August Ferdinand Möbius, Über eine besondere Art von Umkehrung der Reihen. Journal für die reine und angewandte Mathematik 9 (1832), 105-123.
FORMULA
T(n, k) = M(n, k)^(-1), where M(n, k) = [k <= n and k divides n], '(-1)' denotes matrix inversion, and '[ ]' denotes the Iverson bracket.
T(n, k) = k^n if k = 0, otherwise Moebius(n/k) if k divides n, otherwise 0.
Sum_{k=0..n} k*T(n, k) = phi(n) = A000010(n).
EXAMPLE
Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, -1, 1;
[3] 0, -1, 0, 1;
[4] 0, 0, -1, 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, 0, 0, 0, -1, 0, 0, 0, 1;
[9] 0, 0, 0, -1, 0, 0, 0, 0, 0, 1;
MAPLE
A363914 := (n, k) -> ifelse(k = 0, k^n, ifelse(irem(n, k) = 0, NumberTheory:-mu(n/k), 0)): for n from 0 to 9 do seq(A363914(n, k), k = 0..n) od;
# By inverting the Moebius matrix:
divides := (k, n) -> ifelse(k > n, 0, ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0)): M := Matrix(10, (n, k) -> divides(k - 1, n - 1)):
# The shift in k, n is necessary because Maple's 'Matrix' is (1, 1)-based.
LinearAlgebra:-MatrixInverse(M);
PROG
(SageMath)
M = matrix(ZZ, 10, 10, lambda n, k: k <= n and ZZ(k).divides(ZZ(n)))
M.inverse()
# Alternative:
def A363914(n, k):
if k == 0: return k^n
if k.divides(n): return moebius(n // k)
return 0
for n in range(10): print([A363914(n, k) for k in srange(n + 1)])
CROSSREFS
Variant: A054525 (subtriangle with offset (1,1)).
Cf. A113704 (inverse matrix), A008683 (column 1), A019590 (row sums (up to shift)), A130779 (alternating row sums (up to sign)), A000010.
Sequence in context: A318962 A128430 A176330 * A266246 A177990 A287722
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jul 01 2023
EXTENSIONS
Name edited by Peter Luschny, Jul 29 2023
STATUS
approved