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 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. Sequence in context: A126089 A111848 A204384 * A260043 A185057 A343720 Adjacent sequences:  A102409 A102410 A102411 * A102413 A102414 A102415 KEYWORD sign,tabf,uned AUTHOR André F. Labossière, Jan 07 2005 STATUS approved

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Last modified May 9 05:01 EDT 2021. Contains 343688 sequences. (Running on oeis4.)