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%I #19 Jun 16 2016 23:27:27
%S 0,1,0,-16,5,1,0,5256,-3068,276,32,0,2070720,2367420,-912150,53220,
%T 3510,0,-36031524480,15327895296,-40587120,-387492840,21414120,758184,
%U 840,-212459319878400,-75473246681280,38182549456800,-2562251680800,-195611371200,13639812480,285616800,453600
%N 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)!.
%C 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.
%e Triangle starts:
%e 0, 1, 0;
%e -16, 5, 1, 0;
%e 5256, -3068, 276, 32, 0;
%e 2070720, 2367420, -912150, 53220, 3510, 0;
%e -36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840;
%e ...
%e !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 => !11 = [ -212459319878400 -75473246681280*11 +38182549456800*11^2 -2562251680800*11^3 -195611371200*11^4 +13639812480*11^5 +285616800*11^6 +453600*11^7 ]/10! = 4037914.
%Y Cf. A102412, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.
%K sign,tabf,uned
%O 1,4
%A _André F. Labossière_, Jan 07 2005
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