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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)!.

%I A102412 #15 Jun 16 2016 23:27:27
%S A102412 0,1,-4,4,0,96,-396,108,0,1012320,-192900,-64890,11460,90,-2038014720,
%T A102412 1977810240,-304486560,-12131280,2792160,21840,-33190735737600,
%U A102412 4445760574080,2334485260800,-394554283200,2330344800,1198048320,8215200
%N 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)!.
%C A102412 Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.
%e A102412 Triangle starts:
%e A102412 0, 1;
%e A102412 -4, 4, 0;
%e A102412 96, -396, 108, 0;
%e A102412 1012320, -192900, -64890, 11460, 90;
%e A102412 -2038014720, 1977810240, -304486560, -12131280, 2792160, 21840;
%e A102412 ...
%e A102412 !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.
%e A102412 => !11 = [ -33190735737600 +4445760574080*11 +2334485260800*11^2 -394554283200*11^3 +2330344800*11^4 +1198048320*11^5 +8215200*11^6 ]/10! = 4037914.
%Y A102412 Cf. A102411, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.
%K A102412 sign,tabf,uned
%O A102412 1,3
%A A102412 _André F. Labossière_, Jan 07 2005