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A134530 Matrix log of triangle A111636, where A111636(n,k) = (2^k)^(n-k)*C(n,k) for n>=k>=0.

%I A134530 #3 Mar 30 2012 18:37:07
%S A134530 0,1,0,-1,4,0,5,-12,12,0,-79,160,-96,32,0,3377,-6320,3200,-640,80,0,
%T A134530 -362431,648384,-303360,51200,-3840,192,0,93473345,-162369088,
%U A134530 72619008,-11325440,716800,-21504,448,0,-56272471039,95716705280,-41566486528,6196822016,-362414080,9175040,-114688,1024,0
%N A134530 Matrix log of triangle A111636, where A111636(n,k) = (2^k)^(n-k)*C(n,k) for n>=k>=0.
%F A134530 T(n,k) = A134531(n-k)*(2^k)^(n-k)*C(n,k), where A134531 is column 0 and satisfies: G.f.: Sum_{n>=0} A134531(n)*x^n/[n!*2^(n*(n-1)/2)] = log(Sum_{n>=0}x^n/[n!*2^(n*(n-1)/2)]).
%e A134530 Triangle begins:
%e A134530 0,
%e A134530 1, 0;
%e A134530 -1, 4, 0;
%e A134530 5, -12, 12, 0;
%e A134530 -79, 160, -96, 32, 0;
%e A134530 3377, -6320, 3200, -640, 80, 0;
%e A134530 -362431, 648384, -303360, 51200, -3840, 192, 0;
%e A134530 93473345, -162369088, 72619008, -11325440, 716800, -21504, 448, 0; ...
%e A134530 Matrix exponentiation yields triangle A111636, which begins:
%e A134530 1;
%e A134530 1, 1;
%e A134530 1, 4, 1;
%e A134530 1, 12, 12, 1;
%e A134530 1, 32, 96, 32, 1;
%e A134530 1, 80, 640, 640, 80, 1; ...
%o A134530 (PARI) {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,2^((c-1)*(r-c))*binomial(r-1,c-1))),L); L=sum(i=1,#M,-(M^0-M)^i/i);L[n+1,k+1]}
%Y A134530 Cf. A134531 (column 0); related triangles: A111636, A117401; A011266.
%K A134530 sign,tabl
%O A134530 0,5
%A A134530 _Paul D. Hanna_, Oct 30 2007