A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

1, 1, 1, 3, -1, 1, 2, 3, -2, 1, 4, -1, 4, -3, 1, 4, 3, -5, 7, -4, 1, 6, -3, 10, -13, 11, -5, 1, 4, 8, -14, 23, -24, 16, -6, 1, 7, -2, 15, -33, 46, -40, 22, -7, 1, 7, 4, -15, 47, -79, 86, -62, 29, -8, 1, 9, -6, 30, -73, 131, -166, 148, -91, 37, -9, 1, 7, 12, -37, 103, -204, 297, -314, 239, -128, 46, -10, 1

Offset

1,4

Comments

A128316 * [1,2,3...] = A000034: [1,2,1,2,...].

Links

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

Formula

Sum_{k=1..n} T(n, k) = A000027(n) (row sums).

T(n, 1) = A059851(n).

From G. C. Greubel, Jun 23 2024: (Start)

T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.

T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).

T(2*n-1, n) = (-1)^(n-1)*A026641(n).

T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).

Sum_{k=1..n} k*T(n, k) = A032766(n-1).

Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)

Example

First few rows of the triangle:
  1;
  1,  1;
  3, -1,   1;
  2,  3   -2,   1;
  4, -1,   4,  -3,   1;
  4,  3,  -5,   7,  -4,  1;
  6, -3,  10, -13,  11, -5,  1;
  4,  8, -14,  23, -24, 16, -6, 1;
  ...

Mathematica

T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1, k-1], {j, k, n}];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jun 23 2024 *)

Prog

(Magma)
A128316:= func< n, k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1, k-1): j in [k..n]]) >;
[A128316(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 23 2024
(SageMath)
def A128316(n, k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1, k-1) for j in range(k, n+1))
flatten([[A128316(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Jun 23 2024

Crossrefs

Cf. A000012, A000034, A014300, A026641, A059851, A128315.

Sums include: A000027 (row), A032766, A047215, A344817 (alternating sign).

Sequence in context: A030727 A278564 A269973 * A065836 A078712 A287218

Adjacent sequences: A128313 A128314 A128315 * A128317 A128318 A128319

Keyword

tabl,sign

Author

Gary W. Adamson, Feb 25 2007

Extensions

a(28) = 1 inserted and more terms from Georg Fischer, Jun 06 2023

Status

approved