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A102409 Even 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+3, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) * n^(i-1) / (2*k-2)!.

%I A102409 #16 Jun 16 2016 23:27:26
%S A102409 0,1,0,0,0,-20,8,0,0,20280,-6530,-1275,362,3,0,-8749440,21627600,
%T A102409 -4871940,-66510,48300,390,0,-261763004160,72965762016,13117344800,
%U A102409 -3757930680,72406040,13101144,90440,0,-974260634054400,-1140185248443360,353509119454680,-8136128999880,-3234018579750
%N A102409 Even 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+3, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) * n^(i-1) / (2*k-2)!.
%C A102409 Incidentally, 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 A101751.
%e A102409 Triangle starts:
%e A102409 0, 1, 0, 0;
%e A102409 0, -20, 8, 0, 0;
%e A102409 20280, -6530, -1275, 362, 3, 0;
%e A102409 -8749440, 21627600, -4871940, -66510, 48300, 390, 0;
%e A102409 -261763004160, 72965762016, 13117344800, -3757930680, 72406040, 13101144, 90440, 0;
%e A102409 ...
%e A102409 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!.
%e A102409 => 11! = [ -974260634054400 -1140185248443360*11 +353509119454680*11^2 -8136128999880*11^3 -3234018579750*11^4 +109743298560*11^5 +6053880420*11^6 +34067880*11^7 +9450*11^8 ]/10! = 39916800.
%Y A102409 Cf. A102410, A008276, A094216, A000142, A094638, A101751, A102411, A102412, A101752, A003422, A101559, A101032, A099731.
%K A102409 sign,tabf,uned
%O A102409 1,6
%A A102409 _André F. Labossière_, Jan 07 2005