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A121438 Matrix inverse of triangle A122178, where A122178(n,k) = C( n*(n+1)/2 + n-k - 1, n-k) for n>=k>=0.

%I A121438 #3 Mar 30 2012 18:36:58
%S A121438 1,-1,1,-3,-3,1,-17,-3,-6,1,-160,-25,5,-10,1,-2088,-285,-35,30,-15,1,
%T A121438 -34307,-4179,-420,-91,84,-21,1,-675091,-74823,-6916,-497,-322,182,
%U A121438 -28,1,-15428619,-1577763,-135639,-10080,-63,-1002,342,-36,1,-400928675,-38209725,-3082905,-215700,-14139,2655,-2625
%N A121438 Matrix inverse of triangle A122178, where A122178(n,k) = C( n*(n+1)/2 + n-k - 1, n-k) for n>=k>=0.
%C A121438 A triangle having similar properties and complementary construction is the dual triangle A121434.
%F A121438 T(n,k) = [A121412^(-n*(n+1)/2)](n,k) for n>=k>=0; i.e., row n of A122178^-1 equals row n of matrix power A121412^(-n*(n+1)/2).
%e A121438 Triangle, A122178^-1, begins:
%e A121438 1;
%e A121438 -1, 1;
%e A121438 -3, -3, 1;
%e A121438 -17, -3, -6, 1;
%e A121438 -160, -25, 5, -10, 1;
%e A121438 -2088, -285, -35, 30, -15, 1;
%e A121438 -34307, -4179, -420, -91, 84, -21, 1;
%e A121438 -675091, -74823, -6916, -497, -322, 182, -28, 1;
%e A121438 -15428619, -1577763, -135639, -10080, -63, -1002, 342, -36, 1; ...
%e A121438 Triangle A121412 begins:
%e A121438 1;
%e A121438 1, 1;
%e A121438 3, 1, 1;
%e A121438 18, 4, 1, 1;
%e A121438 170, 30, 5, 1, 1; ...
%e A121438 Row 3 of A122178^-1 equals row 3 of A121412^(-6), which begins:
%e A121438 1;
%e A121438 -6, 1;
%e A121438 3, -6, 1;
%e A121438 -17, -3, -6, 1; ...
%e A121438 Row 4 of A122178^-1 equals row 4 of A121412^(-10), which begins:
%e A121438 1;
%e A121438 -10, 1;
%e A121438 25, -10, 1;
%e A121438 -15, 15, -10, 1;
%e A121438 -160, -25, 5, -10, 1; ...
%o A121438 (PARI) /* Matrix Inverse of A122178 */ {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r*(r-1)/2+r-c-1,r-c)))); return((M^-1)[n+1,k+1])}
%Y A121438 Cf. A122178 (matrix inverse); A121412; variants: A121439, A121440, A121441; A121434 (dual).
%K A121438 sign,tabl
%O A121438 0,4
%A A121438 _Paul D. Hanna_, Aug 29 2006