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A121334 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k, n-k), for n>=k>=0.

%I A121334 #3 Mar 30 2012 18:36:58
%S A121334 1,2,1,10,4,1,84,28,7,1,1001,286,66,11,1,15504,3876,816,136,16,1,
%T A121334 296010,65780,12650,2024,253,22,1,6724520,1344904,237336,35960,4495,
%U A121334 435,29,1,177232627,32224114,5245786,749398,91390,9139,703,37,1,5317936260
%N A121334 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k, n-k), for n>=k>=0.
%C A121334 A triangle having similar properties and complementary construction is the dual triangle A122175.
%F A121334 Remarkably, row n of the matrix inverse (A121439) equals row n of A121412^(-n*(n+1)/2-1). 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 A121334 Triangle begins:
%e A121334 1;
%e A121334 2, 1;
%e A121334 10, 4, 1;
%e A121334 84, 28, 7, 1;
%e A121334 1001, 286, 66, 11, 1;
%e A121334 15504, 3876, 816, 136, 16, 1;
%e A121334 296010, 65780, 12650, 2024, 253, 22, 1;
%e A121334 6724520, 1344904, 237336, 35960, 4495, 435, 29, 1;
%e A121334 177232627, 32224114, 5245786, 749398, 91390, 9139, 703, 37, 1; ...
%o A121334 (PARI) T(n,k)=binomial(n*(n+1)/2+n-k,n-k)
%Y A121334 Cf. A121439 (matrix inverse); A121412; variants: A122178, A121335, A121336; A122175 (dual).
%K A121334 nonn,tabl
%O A121334 0,2
%A A121334 _Paul D. Hanna_, Aug 29 2006