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A121335 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 1, n-k), for n>=k>=0.

%I A121335 #3 Mar 30 2012 18:36:58
%S A121335 1,3,1,15,5,1,120,36,8,1,1365,364,78,12,1,20349,4845,969,153,17,1,
%T A121335 376740,80730,14950,2300,276,23,1,8347680,1623160,278256,40920,4960,
%U A121335 465,30,1,215553195,38320568,6096454,850668,101270,9880,741,38,1,6358402050
%N A121335 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 1, n-k), for n>=k>=0.
%C A121335 A triangle having similar properties and complementary construction is the dual triangle A122176.
%F A121335 Remarkably, row n of the matrix inverse (A121440) equals row n of A121412^(-n*(n+1)/2-2). 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 A121335 Triangle begins:
%e A121335 1;
%e A121335 3, 1;
%e A121335 15, 5, 1;
%e A121335 120, 36, 8, 1;
%e A121335 1365, 364, 78, 12, 1;
%e A121335 20349, 4845, 969, 153, 17, 1;
%e A121335 376740, 80730, 14950, 2300, 276, 23, 1;
%e A121335 8347680, 1623160, 278256, 40920, 4960, 465, 30, 1;
%e A121335 215553195, 38320568, 6096454, 850668, 101270, 9880, 741, 38, 1; ...
%o A121335 (PARI) T(n,k)=binomial(n*(n+1)/2+n-k+1,n-k)
%Y A121335 Cf. A121440 (matrix inverse); A121412; variants: A122178, A121334, A121336; A122176 (dual).
%K A121335 nonn,tabl
%O A121335 0,2
%A A121335 _Paul D. Hanna_, Aug 29 2006