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A327029 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.

%I A327029 #29 Jun 20 2024 09:33:52
%S A327029 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 A327029 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 A327029 0,11,5,10,11,10,7,5,3,2,1,1,0,12,17,19,19,14,12,7,5,3,2,1,1
%N A327029 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 A327029 Dirichlet convolution of phi(n) and A008284(n,k) for n >= 1. - _Richard L. Ollerton_, May 07 2021
%H A327029 Alois P. Heinz, <a href="/A327029/b327029.txt">Rows n = 0..200, flattened</a>
%F A327029 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A327029 For n >= 1, T(n,k) = Sum_{i=1..n} A008284(gcd(n,i),k).
%F A327029 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 A327029 Triangle starts:
%e A327029 [0] [1]
%e A327029 [1] [0, 1]
%e A327029 [2] [0, 2, 1]
%e A327029 [3] [0, 3, 1, 1]
%e A327029 [4] [0, 4, 3, 1, 1]
%e A327029 [5] [0, 5, 2, 2, 1, 1]
%e A327029 [6] [0, 6, 6, 4, 2, 1, 1]
%e A327029 [7] [0, 7, 3, 4, 3, 2, 1, 1]
%e A327029 [8] [0, 8, 8, 6, 6, 3, 2, 1, 1]
%e A327029 [9] [0, 9, 6, 9, 6, 5, 3, 2, 1, 1]
%o A327029 (SageMath)
%o A327029 def DivisorTriangle(f, T, Len, w = None):
%o A327029     D = [[1]]
%o A327029     for n in (1..Len-1):
%o A327029         r = lambda k: [f(d)*T(n//d,k) for d in divisors(n)]
%o A327029         L = [sum(r(k)) for k in (0..n)]
%o A327029         if w != None: L = [*map(lambda v: v * w(n), L)]
%o A327029         D.append(L)
%o A327029     return D
%o A327029 DivisorTriangle(euler_phi, A008284, 10)
%Y A327029 Cf. A008284, A000010, A078392 (row sums), A282750.
%Y A327029 Cf. A000041 (where reversed rows converge to).
%Y A327029 T(2n,n) gives A052810.
%K A327029 nonn,tabl
%O A327029 0,5
%A A327029 _Peter Luschny_, Aug 24 2019