A102411 Even triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.

0, 1, 0, -16, 5, 1, 0, 5256, -3068, 276, 32, 0, 2070720, 2367420, -912150, 53220, 3510, 0, -36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840, -212459319878400, -75473246681280, 38182549456800, -2562251680800, -195611371200, 13639812480, 285616800, 453600

Offset: 1

Author

André F. Labossière, Jan 07 2005

Keywords

• sign
• tabf
• uned

Comments

The sum of signed coefficients for each k-th row is divisible by (2*k-2)!. Moreover, another variant (but an incomplete one, and sorted differently) of the above sequence is presented in A101752.

Examples

			Triangle starts:
0, 1, 0;
-16, 5, 1, 0;
5256, -3068, 276, 32, 0;
2070720, 2367420, -912150, 53220, 3510, 0;
-36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840;
...
!11=4037914; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing !11.
=> !11 = [ -212459319878400 -75473246681280*11 [phone]*11^2 -2562251680800*11^3 -195611371200*11^4 [phone]*11^5 [phone]*11^6 +453600*11^7 ]/10! = 4037914.
		

Crossrefs

Cf. A102412, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.