A121440 - OEIS

%I #3 Mar 30 2012 18:36:58

%S 1,-3,1,0,-5,1,-12,4,-8,1,-129,-22,18,-12,1,-1785,-238,-51,51,-17,1,

%T -30291,-3634,-345,-161,115,-23,1,-608565,-66750,-6111,-285,-505,225,

%U -30,1,-14112744,-1432296,-122227,-9177,665,-1387,399,-38,1,-370746528,-35129022,-2818543,-196037,-14335,4841,-3337,658

%N Matrix inverse of triangle A121335, where A121335(n,k) = C( n*(n+1)/2 + n-k + 1, n-k) for n>=k>=0.

%C A triangle having similar properties and complementary construction is the dual triangle A121436.

%F T(n,k) = [A121412^(-n*(n+1)/2 - 2)](n,k) for n>=k>=0; i.e., row n of A121335^-1 equals row n of matrix power A121412^(-n*(n+1)/2 - 2).

%e Triangle, A121335^-1, begins:

%e 1;

%e -3, 1;

%e 0, -5, 1;

%e -12, 4, -8, 1;

%e -129, -22, 18, -12, 1;

%e -1785, -238, -51, 51, -17, 1;

%e -30291, -3634, -345, -161, 115, -23, 1;

%e -608565, -66750, -6111, -285, -505, 225, -30, 1;

%e -14112744, -1432296, -122227, -9177, 665, -1387, 399, -38, 1; ...

%e Triangle A121412 begins:

%e 1;

%e 1, 1;

%e 3, 1, 1;

%e 18, 4, 1, 1;

%e 170, 30, 5, 1, 1; ...

%e Row 3 of A121335^-1 equals row 3 of A121412^(-8), which begins:

%e 1;

%e -8, 1;

%e 12, -8, 1;

%e -12, 4, -8, 1; ...

%e Row 4 of A121335^-1 equals row 4 of A121412^(-12), which begins:

%e 1;

%e -12, 1;

%e 42, -12, 1;

%e -34, 30, -12, 1;

%e -129, -22, 18, -12, 1; ...

%o (PARI) /* Matrix Inverse of A121335 */ {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 Cf. A121335 (matrix inverse); A121412; variants: A121438, A121439, A121441; A121436 (dual).

%K sign,tabl

%O 0,2

%A _Paul D. Hanna_, Aug 29 2006