A121336 - OEIS

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

%S 1,4,1,21,6,1,165,45,9,1,1820,455,91,13,1,26334,5985,1140,171,18,1,

%T 475020,98280,17550,2600,300,24,1,10295472,1947792,324632,46376,5456,

%U 496,31,1,260932815,45379620,7059052,962598,111930,10660,780,39,1

%N Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 2, n-k), for n>=k>=0.

%C A triangle having similar properties and complementary construction is the dual triangle A122177.

%F Remarkably, row n of the matrix inverse (A121441) equals row n of A121412^(-n*(n+1)/2-3). Further, the following matrix products of triangles of binomial coefficients are equal: A121412 = A121334*A122178^-1 = A121335*A121334^-1 = A121336*A121335^-1, where row n of H=A121412 equals row (n-1) of H^(n+1) with an appended '1'.

%e Triangle begins:

%e 1;

%e 4, 1;

%e 21, 6, 1;

%e 165, 45, 9, 1;

%e 1820, 455, 91, 13, 1;

%e 26334, 5985, 1140, 171, 18, 1;

%e 475020, 98280, 17550, 2600, 300, 24, 1;

%e 10295472, 1947792, 324632, 46376, 5456, 496, 31, 1;

%e 260932815, 45379620, 7059052, 962598, 111930, 10660, 780, 39, 1; ...

%o (PARI) T(n,k)=binomial(n*(n+1)/2+n-k+2,n-k)

%Y Cf. A121441 (matrix inverse); A121412; variants: A122178, A121334, A121335; A122177 (dual).

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Aug 29 2006