A128180 Triangle read by rows: A002260 * A097807 as infinite lower triangular matrices.

1, -1, 2, 2, -1, 3, -2, 3, -1, 4, 3, -2, 4, -1, 5, -3, 4, -2, 5, -1, 6, 4, -3, 5, -2, 6, -1, 7, -4, 5, -3, 6, -2, 7, -1, 8, 5, -4, 6, -3, 7, -2, 8, -1, 9, -5, 6, -4, 7, -3, 8, -2, 9, -1, 10, 6, -5, 7, -4, 8, -3, 9, -2, 10, -1, 11, -6, 7, -5, 8, -4, 9, -3, 10, -2, 11, -1, 12

Offset

1,3

Comments

Unsigned row sums are the triangular numbers, A000217 by virtue of the fact that each row is a permutation of the natural numbers.

A128179 = A097807 * A002260.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Formula

Equals A002260 * A097807 as infinite lower triangular matrices.

From Franklin T. Adams-Watters, Apr 12 2011: (Start)

T(n,k) = (2k - 1 + (-1)^(n-k)*(2n+1))/4.

|T(n,k)| = (2n+1 + (-1)^(n-k)*(2k-1))/4. (End)

From Andrew Howroyd, Sep 21 2025: (Start)

T(n,k) = (-1)^(n+k) * A209279(n,k).

G.f.: x*y*(1 + y*x^2)/((1 - x)*(1 + x)^2*(1 - y*x)^2). (End)

Example

Triangle begins:
   1;
  -1,  2;
   2, -1,  3;
  -2,  3, -1,  4;
   3, -2,  4, -1,  5;
  -3,  4, -2,  5, -1,  6;
   4, -3,  5, -2,  6, -1,  7;
  ...

Prog

(PARI) T(n, k)=(2*k-1+(-1)^(n-k)*(2*n+1))/4 \\ Franklin T. Adams-Watters, Apr 12 2011

Crossrefs

Row sums are A008794.

Cf. A000217, A002260, A097807, A128179, A209279.

Sequence in context: A116608 A002947 A241605 * A209279 A074754 A322529

Adjacent sequences: A128177 A128178 A128179 * A128181 A128182 A128183

Keyword

tabl,sign

Author

Gary W. Adamson, Feb 17 2007

Extensions

a(56) onwards from Andrew Howroyd, Sep 21 2025

Status

approved