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A102410 Odd 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+2, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.

%I A102410 #16 Jun 16 2016 23:27:27
%S A102410 1,0,0,-6,3,1,0,2400,-2024,264,32,0,2570400,909720,-666540,55800,3420,
%T A102410 0,-19071521280,12195884736,-762499920,-282106440,22425480,741384,840,
%U A102410 -219303218534400,-11953192930560,27128332828800,-2808016545600,-125442525600,14164990560,280576800
%N A102410 Odd 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+2, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.
%C A102410 Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.
%e A102410 Triangle starts:
%e A102410 1, 0, 0;
%e A102410 -6, 3, 1, 0;
%e A102410 2400, -2024, 264, 32, 0;
%e A102410 2570400, 909720, -666540, 55800, 3420, 0;
%e A102410 -19071521280, 12195884736, -762499920, -282106440, 22425480, 741384, 840;
%e A102410 ...
%e A102410 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 A102410 => 11! = [ -219303218534400 -11953192930560*11 +27128332828800*11^2 -2808016545600*11^3 -125442525600*11^4 +14164990560*11^5 +280576800*11^6 +453600*11^7 ]/10! = 39916800.
%Y A102410 Cf. A102409, A008276, A094216, A000142, A094638, A101751, A102411, A102412, A101752, A003422, A101559, A101032, A099731.
%K A102410 sign,tabf,uned
%O A102410 1,4
%A A102410 _André F. Labossière_, Jan 07 2005