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A122175 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

%I A122175 #9 Jul 22 2024 23:31:09
%S A122175 1,1,1,1,2,1,1,3,4,1,1,4,10,7,1,1,5,20,28,11,1,1,6,35,84,66,16,1,1,7,
%T A122175 56,210,286,136,22,1,1,8,84,462,1001,816,253,29,1,1,9,120,924,3003,
%U A122175 3876,2024,435,37,1,1,10,165,1716,8008,15504,12650,4495,703,46,1,1,11,220
%N A122175 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.
%C A122175 Remarkably, column k of the matrix inverse (A121435) equals signed column k of matrix power: A107876^(k*(k+1)/2 + 1).
%e A122175 Triangle begins:
%e A122175 1;
%e A122175 1, 1;
%e A122175 1, 2, 1;
%e A122175 1, 3, 4, 1;
%e A122175 1, 4, 10, 7, 1;
%e A122175 1, 5, 20, 28, 11, 1;
%e A122175 1, 6, 35, 84, 66, 16, 1;
%e A122175 1, 7, 56, 210, 286, 136, 22, 1;
%e A122175 1, 8, 84, 462, 1001, 816, 253, 29, 1; ...
%t A122175 Table[Binomial[(k(k+1))/2+n-k,n-k],{n,0,20},{k,0,n}]//Flatten (* _Harvey P. Dale_, Mar 22 2016 *)
%o A122175 (PARI) T(n,k)=if(n<k||k<0,0,binomial(k*(k+1)/2+n-k,n-k))
%Y A122175 Cf. A121435 (inverse); variants: A098568, A122176, A122177.
%K A122175 nonn,tabl
%O A122175 0,5
%A A122175 _Paul D. Hanna_, Aug 25 2006