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%I #4 Mar 30 2012 18:37:02
%S 1,2,1,10,4,1,165,45,7,1,5985,1140,136,11,1,376992,52360,4960,325,16,
%T 1,36288252,3819816,292825,16215,666,22,1,4935847320,406481544,
%U 25621596,1215450,43680,1225,29,1,899749479915,59487568920,3127595016,128164707
%N Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 1, n-k) for n>=k>=0.
%C Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.
%F T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 1, n-k) for n>=k>=0.
%e Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 1, n-k) is illustrated by:
%e T(n=4,k=1) = C( C(6,3) - C(3,3) + 1, n-k) = C(20,3) = 1140;
%e T(n=4,k=2) = C( C(6,3) - C(4,3) + 1, n-k) = C(17,2) = 136;
%e T(n=5,k=2) = C( C(7,3) - C(4,3) + 1, n-k) = C(32,3) = 4960.
%e Triangle begins:
%e 1;
%e 2, 1;
%e 10, 4, 1;
%e 165, 45, 7, 1;
%e 5985, 1140, 136, 11, 1;
%e 376992, 52360, 4960, 325, 16, 1;
%e 36288252, 3819816, 292825, 16215, 666, 22, 1;
%e 4935847320, 406481544, 25621596, 1215450, 43680, 1225, 29, 1; ...
%o (PARI) T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+1, n-k)
%Y Columns: A126451, A126452; A126453 (row sums); variants: A126445, A126454, A126457, A107867.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Dec 27 2006
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