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A118791 Triangle where T(n,k) = -n!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.

%I A118791 #5 Mar 30 2012 18:36:57
%S A118791 1,-1,3,2,-9,19,-6,36,-103,207,24,-180,650,-1605,3211,-120,1080,-4710,
%T A118791 13860,-32191,64383,720,-7560,38640,-132300,351722,-790629,1581259,
%U A118791 -5040,60480,-354480,1386000,-4163166,10433556,-22974463,45948927,40320,-544320,3598560,-15830640,53117064
%N A118791 Triangle where T(n,k) = -n!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.
%C A118791 [0, diagonal] = A052886 with e.g.f.: (1-sqrt(5-4*exp(x)))/2. [0, row sums] = A117271 with e.g.f.: log((3-sqrt(5-4*exp(x)))/2). [0, unsigned row sums] = A118792 with e.g.f.: -log((1+sqrt(5-4*exp(x)))/2). Here [0, sequence] indicates that the sequence is offset with a leading zero.
%e A118791 Triangle begins:
%e A118791 1;
%e A118791 -1, 3;
%e A118791 2,-9, 19;
%e A118791 -6, 36,-103, 207;
%e A118791 24,-180, 650,-1605, 3211;
%e A118791 -120, 1080,-4710, 13860,-32191, 64383;
%e A118791 720,-7560, 38640,-132300, 351722,-790629, 1581259;
%e A118791 -5040, 60480,-354480, 1386000,-4163166, 10433556,-22974463, 45948927;
%e A118791 which is formed from the powers of F(x) = x/log(1-x-x^2):
%e A118791 F(x)^1 = (-1) + 3/2*x - 11/12*x^2 + 9/8*x^3 - 641/720*x^4 +...
%e A118791 F(x)^2 = ( 1 - 3*x)/1! + 49/12*x^2 - 5*x^3 + 1439/240*x^4 +...
%e A118791 F(x)^3 = (-2 + 9*x - 19*x^2)/2! + 15*x^3 - 5161/240*x^4 +...
%e A118791 F(x)^4 = ( 6 - 36*x + 103*x^2 - 207*x^3)/3! + 42239/720*x^4 +...
%e A118791 F(x)^5 = (-24 + 180*x - 650*x^2 + 1605*x^3 - 3211*x^4)/4! +...
%o A118791 (PARI) {T(n,k)=local(x=X+X^2*O(X^(k+2)));-n!*polcoeff(((x/log(1-x-x^2)))^(n+1),k,X)}
%Y A118791 Cf. A052886 (diagonal), A117271 (row sums), A118792 (unsigned row sums); A118793 (variant).
%K A118791 sign,tabl
%O A118791 0,3
%A A118791 _Paul D. Hanna_, Apr 30 2006