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A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

%I A129479 #13 Apr 07 2025 13:31:53
%S A129479 1,2,1,2,1,1,3,1,1,1,4,0,0,1,1,4,3,1,0,1,1,6,0,0,0,0,1,1,6,2,1,1,0,0,
%T A129479 1,1,6,2,2,0,0,0,0,1,1,8,4,0,1,1,0,0,0,1,1,10,0,0,0,0,0,0,0,0,1,1,6,4,
%U A129479 4,2,1,1,0,0,0,0,1,1,12,0,0,0,0,0,0,0,0,0,0,1,1
%N A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.
%H A129479 G. C. Greubel, <a href="/A129479/b129479.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A129479 Sum_{k=1..n} T(n, k) = A053158(n) (row sums).
%F A129479 T(n, 1) = A126246(n).
%F A129479 From _G. C. Greubel_, Feb 11 2024: (Start)
%F A129479 T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.
%F A129479 T(2*n-1, n) = A019590(n).
%F A129479 T(2*n, n) = A054977(n).
%F A129479 T(2*n+1, n) = A000038(n).
%F A129479 T(3*n, n) = A063524(n-1).
%F A129479 T(3*n-2, n) = A183918(n+2).
%F A129479 Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)
%e A129479 First few rows of the triangle:
%e A129479   1;
%e A129479   2, 1;
%e A129479   2, 1, 1;
%e A129479   3, 1, 1, 1;
%e A129479   4, 0, 0, 1, 1;
%e A129479   4, 3, 1, 0, 1, 1;
%e A129479   6, 0, 0, 0, 0, 1, 1;
%e A129479   6, 2, 1, 1, 0, 0, 1, 1;
%e A129479   ...
%t A129479 A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];
%t A129479 T[n_, k_]:= If[k<n, Sum[A054523[n, j+k], {j,0,1}], 1];
%t A129479 Table[T[n,k],{n,16},{k,n}]//Flatten (* _G. C. Greubel_, Feb 11 2024 *)
%o A129479 (Magma)
%o A129479 A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
%o A129479 A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;
%o A129479 [A129479(n,k): k in [1..n], n in [1..16]]; // _G. C. Greubel_, Feb 11 2024
%o A129479 (SageMath)
%o A129479 def A054523(n,k):
%o A129479     if (k==n): return 1
%o A129479     elif (n%k): return 0
%o A129479     else: return euler_phi(n//k)
%o A129479 def A129479(n, k):
%o A129479     if k<0 or k>n: return 0
%o A129479     elif k==n: return 1
%o A129479     else: return A054523(n,k) + A054523(n,k+1)
%o A129479 flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # _G. C. Greubel_, Feb 11 2024
%Y A129479 Cf. A000010 (alternating row sums), A053158 (row sums).
%Y A129479 Cf. A000038, A019590, A054523, A054977.
%Y A129479 Cf. A063524, A097806, A126246, A183918.
%K A129479 nonn,tabl
%O A129479 1,2
%A A129479 _Gary W. Adamson_, Apr 17 2007