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%I #4 Mar 30 2012 18:36:57
%S 1,-1,3,1,-9,19,-1,18,-103,207,1,-30,325,-1605,3211,-1,45,-785,6930,
%T -32191,64383,1,-63,1610,-22050,175861,-790629,1581259,-1,84,-2954,
%U 57750,-693861,5216778,-22974463,45948927,1,-108,4998,-131922,2213211,-24542910,177555925,-770820885,1541641771
%N Triangle where T(n,k) = -n!/(n-k)!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.
%C [0, diagonal] = A052886 with e.g.f.: (1-sqrt(5-4*exp(x)))/2. [0, row sums] = A118794 with e.g.f.: 1-exp((-1+sqrt(5-4*exp(x)))/2). [0, unsigned row sums] = A118795 with e.g.f.: -1+exp((1-sqrt(5-4*exp(x)))/2). Here [0, sequence] indicates that the sequence is to be offset with leading zero.
%e Triangle begins:
%e 1;
%e -1, 3;
%e 1,-9, 19;
%e -1, 18,-103, 207;
%e 1,-30, 325,-1605, 3211;
%e -1, 45,-785, 6930,-32191, 64383;
%e 1,-63, 1610,-22050, 175861,-790629, 1581259;
%e -1, 84,-2954, 57750,-693861, 5216778,-22974463, 45948927; ...
%e which is formed from the powers of F(x) = x/log(1-x-x^2):
%e F(x)^1 = (-1) + 3/2*x - 11/12*x^2 + 9/8*x^3 - 641/720*x^4 +...
%e F(x)^2 = ( 1 - 3*x) + 49/12*x^2 - 5*x^3 + 1439/240*x^4 +...
%e F(x)^3 = (-1 + 9/2*x - 19/2*x^2) + 15*x^3 - 5161/240*x^4 +...
%e F(x)^4 = ( 1 - 18/3*x + 103/6*x^2 - 207/6*x^3) + 42239/720*x^4 +...
%e F(x)^5 = (-1 + 30/4*x - 325/12*x^2 + 1605/24*x^3 - 3211/24*x^4) +...
%o (PARI) {T(n,k)=local(x=X+X^2*O(X^(k+2)));-n!/(n-k)!*polcoeff(((x/log(1-x-x^2)))^(n+1),k,X)}
%Y Cf. A052886 (diagonal), A118794 (row sums), A118795 (unsigned row sums); A118791 (variant).
%K sign,tabl
%O 0,3
%A _Paul D. Hanna_, Apr 30 2006
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