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A128317
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Triangle read by rows: T = A054523 * A130595, as a lower triangular matrix.
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1
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1, 0, 1, 3, -2, 1, 0, 4, -3, 1, 5, -4, 6, -4, 1, 0, 5, -9, 10, -5, 1, 7, -6, 15, -20, 15, -6, 1, 0, 12, -24, 36, -35, 21, -7, 1, 9, -12, 30, -56, 70, -56, 28, -8, 1, 0, 9, -30, 80, -125, 126, -84, 36, -9, 1, 11, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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LINKS
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FORMULA
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Equals A054523 * signed A007318 as infinite lower triangular matrices. A007318 is signed by columns: (+, -, +, ...).
Sum_{k=1..n} T(n, k) = A000010(n) (row sums).
T(n, k) = Sum_{j=k..n} (-1)^(k+j)*A054523(n,j)*binomial(j-1, k-1).
T(n, k) = Sum_{d|n} (-1)^(d+k)*EulerPhi(n/d)*binomial(d-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
T(2*n-2, n-1) = (-1)^n*A001700(n-2), n >= 2.
Sum_{k=1..n} k*T(n, k) = A126246(n). (End)
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EXAMPLE
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First few rows of the triangle:
1;
0, 1;
3, -2, 1;
0, 4, -3, 1;
5, -4, 6, -4, 1;
0, 5, -9, 10, -5, 1;
7, -6, 15, -20, 15, -6, 1;
...
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MATHEMATICA
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A128317[n_, k_]:= DivisorSum[n, (-1)^(#+k)*EulerPhi[n/#]*Binomial[#-1, k-1] &];
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PROG
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(Magma)
A128317:= func< n, k | (&+[(-1)^(d+k)*EulerPhi(Floor(n/d))*Binomial(d-1, k-1) : d in Divisors(n)]) >;
(SageMath)
def A128317(n, k): return sum((-1)^(k+j)*euler_phi(n/j)*binomial(j-1, k-1) for j in (1..n) if (j).divides(n))
flatten([[A128317(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Jun 24 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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