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A122178 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k - 1, n-k), for n>=k>=0.

%I A122178 #3 Mar 30 2012 18:36:59
%S A122178 1,1,1,6,3,1,56,21,6,1,715,220,55,10,1,11628,3060,680,120,15,1,230230,
%T A122178 53130,10626,1771,231,21,1,5379616,1107568,201376,31465,4060,406,28,1,
%U A122178 145008513,26978328,4496388,658008,82251,8436,666,36,1,4431613550
%N A122178 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k - 1, n-k), for n>=k>=0.
%C A122178 A triangle having similar properties and complementary construction is the dual triangle A098568.
%F A122178 Remarkably, row n of the matrix inverse (A121438) equals row n of A121412^(-n*(n+1)/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 A122178 Triangle begins:
%e A122178 1;
%e A122178 1, 1;
%e A122178 6, 3, 1;
%e A122178 56, 21, 6, 1;
%e A122178 715, 220, 55, 10, 1;
%e A122178 11628, 3060, 680, 120, 15, 1;
%e A122178 230230, 53130, 10626, 1771, 231, 21, 1;
%e A122178 5379616, 1107568, 201376, 31465, 4060, 406, 28, 1;
%e A122178 145008513, 26978328, 4496388, 658008, 82251, 8436, 666, 36, 1; ...
%o A122178 (PARI) T(n,k)=binomial(n*(n+1)/2+n-k-1,n-k)
%Y A122178 Cf. A121438 (matrix inverse); A121412; variants: A121334, A121335, A121336; A098568 (dual).
%K A122178 nonn,tabl
%O A122178 0,4
%A A122178 _Paul D. Hanna_, Aug 29 2006