OFFSET
0,5
COMMENTS
Dirichlet convolution of phi(n) and A008284(n,k) for n >= 1. - Richard L. Ollerton, May 07 2021
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
From Richard L. Ollerton, May 07 2021: (Start)
For n >= 1, T(n,k) = Sum_{i=1..n} A008284(gcd(n,i),k).
For n >= 1, T(n,k) = Sum_{i=1..n} A008284(n/gcd(n,i),k)*phi(gcd(n,i))/phi(n/gcd(n,i)). (End)
EXAMPLE
Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 2, 1]
[3] [0, 3, 1, 1]
[4] [0, 4, 3, 1, 1]
[5] [0, 5, 2, 2, 1, 1]
[6] [0, 6, 6, 4, 2, 1, 1]
[7] [0, 7, 3, 4, 3, 2, 1, 1]
[8] [0, 8, 8, 6, 6, 3, 2, 1, 1]
[9] [0, 9, 6, 9, 6, 5, 3, 2, 1, 1]
PROG
(SageMath)
def DivisorTriangle(f, T, Len, w = None):
D = [[1]]
for n in (1..Len-1):
r = lambda k: [f(d)*T(n//d, k) for d in divisors(n)]
L = [sum(r(k)) for k in (0..n)]
if w != None: L = [*map(lambda v: v * w(n), L)]
D.append(L)
return D
DivisorTriangle(euler_phi, A008284, 10)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Aug 24 2019
STATUS
approved