A158471 Stirling-like triangle by rows generated from (x-1)*(x-1)*(x-2)*(x-3)*(x-4)*...

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -7, 17, -17, 6, 1, -11, 45, -85, 74, -24, 1, -16, 100, -310, 499, -394, 120, 1, -22, 196, -910, 2359, -3388, 2484, -720, 1, -29, 350, -2282, 8729, -19901, 26200, -18108, 5040, 1, -37, 582, -5082, 26985, -89733, 185408

Offset: 0

Author

Gary W. Adamson, Mar 20 2009

Keywords

• tabl
• sign

Comments

Row sums of the unsigned triangle = A098558: (1, 2, 4, 12, 48, 240, 1440, 10080, ...).

Examples

			First few rows of the unsigned triangle =
1;
1, 1;
1, 2, 1;
1, 4, 5, 2;
1, 7, 17, 17, 6;
1, 11, 45, 85, 74, 24;
1, 16, 100, 310, 499, 394, 120;
1, 22, 196, 910, 2359, 3388, 2484, 720;
1, 29, 350, 2282, 8729, 19901, 26200, 18108, 5040;
1, 37, 582, 5082, 26985, 89733, 185408, 227708, 149904, 40320;
1, 46, 915, 10320, 72723, 332598, 993005, 1896380, 2199276, 1389456, 362880;
...
Example: Row 5 = x^5 - 11x^4 + 45x^3 -85x^2 + 74x - 24 =
(x-1)*(x-1)*(x-2)*(x-3)*(x-4).
		

Crossrefs

Cf. A098558.

Cf. A103718. - R. J. Mathar, Mar 20 2009

Formula

Triangle read by rows, n-th row = n-th degree polynomial with alternating signs generated from n terms of a*b*c*d*...; where a,b,c,... = (x-1), (x-1), (x-2), (x-3), (x-4), ... n-th row, n > 0 = charpoly of an n X n matrix with (1,1,2,3,4,...) in the diagonal and the rest zeros.