A120476 Triangle read by rows: a(n,m)=(2*n-1)*(n-m)*(n+m+1)/2, where n is the column and m the row index.

1, 3, -2, 6, -5, -9, 10, -9, -21, -20, 15, -14, -36, -45, -35, 21, -20, -54, -75, -77, -54, 28, -27, -75, -110, -126, -117, -77, 36, -35, -99, -150, -182, -189, -165, -104, 45, -44, -126, -195, -245, -270, -264, -221, -135, 55, -54, -156, -245, -315, -360, -374, -351, -285, -170

Offset: 0

Author

Roger L. Bagula, Jul 19 2006

Keywords

• sign
• tabl

Comments

Triangular array based on recurrence in Laplace function in J. W. S. Rayleigh.

Examples

			1,
3, -2,
6, -5, -9,
10, -9, -21,-20,
15, -14,-36,-45, -35
		

References

• J. W. S. Rayleigh, The Theory of Sound, volume 2, page 237,Dover, New York,1945

Crossrefs

Cf. A006472.

Programs

Mathematica

a = Table[Table[(m + 1)*(2*n - 1)*(n - m)*(n + m + 1)/(2*(m + 1)), {n, 0, m - 1}], {m, 1, 10}] Flatten[a]

Formula

a(n,m) = (2n-1)*[A000217(n)-A000217(m)] = (1-2n)*A049777(n,m) . - R. J. Mathar, Dec 05 2007

Row sums: sum_{n=0..m-1} a(n,m) = -m(m+1)(3m^2-5m-4)/12. [From R. J. Mathar, Jan 15 2009]

Extensions

Edited by N. J. A. Sloane, Oct 01 2006