A165507 Triangle T(n,m) read by rows: numerator of 1/(1+n-m)^2 - 1/m^2.

0, -3, 3, -8, 0, 8, -15, -5, 5, 15, -24, -3, 0, 3, 24, -35, -21, -7, 7, 21, 35, -48, -2, -16, 0, 16, 2, 48, -63, -45, -1, -9, 9, 1, 45, 63, -80, -15, -40, -5, 0, 5, 40, 15, 80, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, -120, -6, -8, -3, -24, 0, 24, 3, 8, 6, 120

Offset: 1

Author

Paul Curtz, Sep 21 2009

Keywords

• sign
• frac
• tabl
• easy

Comments

The triangle is obtained from the infinite array shown in the comment in A172370 by starting in column 1 and reading diagonally upwards along increasing columns or starting in column -1 and reading diagonally upwards along decreasing columns.

Equivalence of these two interpretations follows from the mirror symmetry m <-> -m along column m=0 in that array.

T(n,m) is antisymmetric (changes sign) with respect to a central zero if the row index n is odd, and with respect to the separator in the middle of the row if the row index n is even: T(n,m) = -T(n,n+1-m).

An appropriate triangle of denominators is in A143183.

Examples

			The triangle starts in row n=1 with columns 1<=m<=n as
0;
-3,3;
-8,0,8;
-15,-5,5,15;
-24,-3,0,3,24;
-35,-21,-7,7,21,35;
-48,-2,-16,0,16,2,48;
		

Links

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

Crossrefs

Cf. A120072, A143183, A165441.

Programs

Magma

[[Numerator(1/(n-k+1)^2 - 1/k^2): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Oct 21 2018

Maple

A165507 := proc(n,m) 1/(1+n-m)^2-1/m^2 ; numer(%) ; end proc:

Mathematica

Table[Numerator[1/(n-k+1)^2 - 1/k^2], {n,1,15}, {k,1,n}]//Flatten (* G. C. Greubel, Oct 21 2018 *)

PARI

for(n=1, 15, for(k=1, n, print1(numerator(1/(n-k+1)^2 - 1/k^2), ", "))) \\ G. C. Greubel, Oct 21 2018

Formula

T(n,m) = A173651(1+n,m), m>=1.