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A126457 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 3, n-k) for n>=k>=0.

%I A126457 #4 Mar 30 2012 18:37:02
%S A126457 1,4,1,21,6,1,286,66,9,1,8855,1540,171,13,1,501942,66045,5984,378,18,
%T A126457 1,45057474,4582116,341055,18424,741,24,1,5843355957,470155077,
%U A126457 29034396,1353275,47905,1326,31,1,1029873432159,66983637864,3470108187,140364532
%N A126457 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 3, n-k) for n>=k>=0.
%C A126457 Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.
%F A126457 T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 3, n-k) for n>=k>=0.
%e A126457 Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 3, n-k) is illustrated by:
%e A126457 T(n=4,k=1) = C( C(6,3) - C(3,3) + 3, n-k) = C(22,3) = 1540;
%e A126457 T(n=4,k=2) = C( C(6,3) - C(4,3) + 3, n-k) = C(19,2) = 171;
%e A126457 T(n=5,k=2) = C( C(7,3) - C(4,3) + 3, n-k) = C(34,3) = 5984.
%e A126457 Triangle begins:
%e A126457 1;
%e A126457 4, 1;
%e A126457 21, 6, 1;
%e A126457 286, 66, 9, 1;
%e A126457 8855, 1540, 171, 13, 1;
%e A126457 501942, 66045, 5984, 378, 18, 1;
%e A126457 45057474, 4582116, 341055, 18424, 741, 24, 1;
%e A126457 5843355957, 470155077, 29034396, 1353275, 47905, 1326, 31, 1; ...
%o A126457 (PARI) T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+3, n-k)
%Y A126457 Columns: A126458, A126459; variants: A126445, A126450, A126454, A107873.
%K A126457 nonn,tabl
%O A126457 0,2
%A A126457 _Paul D. Hanna_, Dec 27 2006