|
|
A102412
|
|
Odd 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+1, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) * n^(i-1) / (2*k-2)!.
|
|
5
|
|
|
0, 1, -4, 4, 0, 96, -396, 108, 0, 1012320, -192900, -64890, 11460, 90, -2038014720, 1977810240, -304486560, -12131280, 2792160, 21840, -33190735737600, 4445760574080, 2334485260800, -394554283200, 2330344800, 1198048320, 8215200
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.
|
|
LINKS
|
|
|
EXAMPLE
|
Triangle starts:
0, 1;
-4, 4, 0;
96, -396, 108, 0;
1012320, -192900, -64890, 11460, 90;
-2038014720, 1977810240, -304486560, -12131280, 2792160, 21840;
...
!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 = [ -33190735737600 +4445760574080*11 +2334485260800*11^2 -394554283200*11^3 +2330344800*11^4 +1198048320*11^5 +8215200*11^6 ]/10! = 4037914.
|
|
CROSSREFS
|
Cf. A102411, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.
|
|
KEYWORD
|
sign,tabf,uned
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|