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A126454 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) for n>=k>=0.

%I A126454 #6 Dec 17 2020 18:35:26
%S A126454 1,3,1,15,5,1,220,55,8,1,7315,1330,153,12,1,435897,58905,5456,351,17,
%T A126454 1,40475358,4187106,316251,17296,703,23,1,5373200880,437353560,
%U A126454 27285336,1282975,45760,1275,30,1,962889794295,63140314380,3295144749,134153712
%N A126454 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) for n>=k>=0.
%C A126454 Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.
%F A126454 T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 2, n-k) for n>=k>=0.
%e A126454 Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) is illustrated by:
%e A126454 T(n=4,k=1) = C( C(6,3) - C(3,3) + 2, n-k) = C(21,3) = 1330;
%e A126454 T(n=4,k=2) = C( C(6,3) - C(4,3) + 2, n-k) = C(18,2) = 153;
%e A126454 T(n=5,k=2) = C( C(7,3) - C(4,3) + 2, n-k) = C(33,3) = 5456.
%e A126454 Triangle begins:
%e A126454 1;
%e A126454 3, 1;
%e A126454 15, 5, 1;
%e A126454 220, 55, 8, 1;
%e A126454 7315, 1330, 153, 12, 1;
%e A126454 435897, 58905, 5456, 351, 17, 1;
%e A126454 40475358, 4187106, 316251, 17296, 703, 23, 1;
%e A126454 5373200880, 437353560, 27285336, 1282975, 45760, 1275, 30, 1; ...
%t A126454 Table[Binomial[Binomial[n+2,3]-Binomial[k+2,3]+2,n-k],{n,0,10},{k,0,n}]// Flatten (* _Harvey P. Dale_, Dec 17 2020 *)
%o A126454 (PARI) T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+2, n-k)
%Y A126454 Columns: A126455, A126456; variants: A126445, A126450, A126457, A107870.
%K A126454 nonn,tabl
%O A126454 0,2
%A A126454 _Paul D. Hanna_, Dec 27 2006