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A129479
Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices..
2
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, 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, 4, 2, 1, 1, 0, 0, 0, 0, 1, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1
OFFSET
1,2
FORMULA
Sum_{k=1..n} T(n, k) = A053158(n) (row sums).
T(n, 1) = A126246(n).
From G. C. Greubel, Feb 11 2024: (Start)
T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.
T(2*n-1, n) = A019590(n).
T(2*n, n) = A054977(n).
T(2*n+1, n) = A000038(n).
T(3*n, n) = A063524(n-1).
T(3*n-2, n) = A183918(n+2).
Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)
EXAMPLE
First few rows of the triangle:
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, 1, 1;
...
MATHEMATICA
A054523[n_, k_]:= If[n==1, 1, If[Divisible[n, k], EulerPhi[n/k], 0]];
T[n_, k_]:= If[k<n, Sum[A054523[n, j+k], {j, 0, 1}], 1];
Table[T[n, k], {n, 16}, {k, n}]//Flatten (* G. C. Greubel, Feb 11 2024 *)
PROG
(Magma)
A054523:= func< n, k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
A129479:= func< n, k | k le n-1 select A054523(n, k) + A054523(n, k+1) else 1 >;
[A129479(n, k): k in [1..n], n in [1..16]]; // G. C. Greubel, Feb 11 2024
(SageMath)
def A054523(n, k):
if (k==n): return 1
elif (n%k): return 0
else: return euler_phi(n//k)
def A129479(n, k):
if k<0 or k>n: return 0
elif k==n: return 1
else: return A054523(n, k) + A054523(n, k+1)
flatten([[A129479(n, k) for k in range(1, n+1)] for n in range(1, 17)]) # G. C. Greubel, Feb 11 2024
CROSSREFS
Cf. A000010 (alternating row sums), A053158 (row sums).
Sequence in context: A084610 A303336 A361462 * A261095 A249809 A075104
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Apr 17 2007
STATUS
approved