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%I #20 May 08 2021 07:46:33
%S 1,0,1,0,2,1,0,3,1,1,0,4,3,1,1,0,5,2,2,1,1,0,6,6,4,2,1,1,0,7,3,4,3,2,
%T 1,1,0,8,8,6,6,3,2,1,1,0,9,6,9,6,5,3,2,1,1,0,10,11,10,10,8,5,3,2,1,1,
%U 0,11,5,10,11,10,7,5,3,2,1,1
%N T(n, k) = Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. Triangle read by rows for 0 <= k <= n.
%C Dirichlet convolution of phi(n) and A008284(n,k) for n >= 1. - _Richard L. Ollerton_, May 07 2021
%F From _Richard L. Ollerton_, May 07 2021: (Start)
%F For n >= 1, T(n,k) = Sum_{i=1..n} A008284(gcd(n,i),k).
%F 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)
%e Triangle starts:
%e [0] [1]
%e [1] [0, 1]
%e [2] [0, 2, 1]
%e [3] [0, 3, 1, 1]
%e [4] [0, 4, 3, 1, 1]
%e [5] [0, 5, 2, 2, 1, 1]
%e [6] [0, 6, 6, 4, 2, 1, 1]
%e [7] [0, 7, 3, 4, 3, 2, 1, 1]
%e [8] [0, 8, 8, 6, 6, 3, 2, 1, 1]
%e [9] [0, 9, 6, 9, 6, 5, 3, 2, 1, 1]
%o (SageMath)
%o def DivisorTriangle(f, T, Len, w = None):
%o D = [[1]]
%o for n in (1..Len-1):
%o r = lambda k: [f(d)*T(n//d,k) for d in divisors(n)]
%o L = [sum(r(k)) for k in (0..n)]
%o if w != None: L = [*map(lambda v: v * w(n), L)]
%o D.append(L)
%o return D
%o DivisorTriangle(euler_phi, A008284, 10)
%Y Cf. A008284, A000010, A078392 (row sums), A282750.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Aug 24 2019
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