A102410 Odd triangle n!. This table read by rows gives the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.

1, 0, 0, -6, 3, 1, 0, 2400, -2024, 264, 32, 0, 2570400, 909720, -666540, 55800, 3420, 0, -19071521280, 12195884736, -762499920, -282106440, 22425480, 741384, 840, -219303218534400, -11953192930560, 27128332828800, -2808016545600, -125442525600, 14164990560, 280576800

Offset: 1

Author

André F. Labossière, Jan 07 2005

Keywords

• sign
• tabf
• uned

Comments

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.

Examples

			Triangle starts:
1, 0, 0;
-6, 3, 1, 0;
2400, -2024, 264, 32, 0;
2570400, 909720, -666540, 55800, 3420, 0;
-19071521280, 12195884736, -762499920, -282106440, 22425480, 741384, 840;
...
11!=39916800; 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! = [ -219303218534400 -11953192930560*11 [phone]*11^2 -2808016545600*11^3 -125442525600*11^4 [phone]*11^5 [phone]*11^6 +453600*11^7 ]/10! = 39916800.
		

Crossrefs

Cf. A102409, A008276, A094216, A000142, A094638, A101751, A102411, A102412, A101752, A003422, A101559, A101032, A099731.