A128317 Triangle read by rows: T = A054523 * A130595, as a lower triangular matrix.

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

Offset: 1

Author

Gary W. Adamson, Feb 25 2007

Keywords

• tabl
• sign

Examples

			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;
  ...
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A000984, A001700, A007318, A054523, A130595.

Sums include: A000010 (row sums), A126246.

Programs

Magma

A128317:= func< n,k | (&+[(-1)^(d+k)*EulerPhi(Floor(n/d))*Binomial(d-1, k-1) : d in Divisors(n)]) >;
[A128317(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024

Mathematica

A128317[n_, k_]:= DivisorSum[n, (-1)^(#+k)*EulerPhi[n/#]*Binomial[#-1, k-1]  &];
Table[A128317[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 24 2024 *)

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

Formula

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).

From G. C. Greubel, Jun 24 2024: (Start)

T(n, k) = A054523 * A130595, as a lower triangular matrix.

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)