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A121336 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 2, n-k), for n>=k>=0.

%I A121336 #3 Mar 30 2012 18:36:58
%S A121336 1,4,1,21,6,1,165,45,9,1,1820,455,91,13,1,26334,5985,1140,171,18,1,
%T A121336 475020,98280,17550,2600,300,24,1,10295472,1947792,324632,46376,5456,
%U A121336 496,31,1,260932815,45379620,7059052,962598,111930,10660,780,39,1
%N A121336 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 2, n-k), for n>=k>=0.
%C A121336 A triangle having similar properties and complementary construction is the dual triangle A122177.
%F A121336 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 A121336 Triangle begins:
%e A121336 1;
%e A121336 4, 1;
%e A121336 21, 6, 1;
%e A121336 165, 45, 9, 1;
%e A121336 1820, 455, 91, 13, 1;
%e A121336 26334, 5985, 1140, 171, 18, 1;
%e A121336 475020, 98280, 17550, 2600, 300, 24, 1;
%e A121336 10295472, 1947792, 324632, 46376, 5456, 496, 31, 1;
%e A121336 260932815, 45379620, 7059052, 962598, 111930, 10660, 780, 39, 1; ...
%o A121336 (PARI) T(n,k)=binomial(n*(n+1)/2+n-k+2,n-k)
%Y A121336 Cf. A121441 (matrix inverse); A121412; variants: A122178, A121334, A121335; A122177 (dual).
%K A121336 nonn,tabl
%O A121336 0,2
%A A121336 _Paul D. Hanna_, Aug 29 2006