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A126465 Triangle T, read by rows, where row n equals row (n-1) of matrix power T^(n(n+1)/2) concatenated with a trailing '1', for n>0, with T(0,0) = 1.

%I A126465 #3 Mar 30 2012 18:37:02
%S A126465 1,1,1,3,1,1,33,6,1,1,855,105,10,1,1,40475,3710,255,15,1,1,3039204,
%T A126465 219625,11935,525,21,1,1,331630320,19545316,879571,31584,966,28,1,1,
%U A126465 49563943161,2437990653,93365328,2856819,72786,1638,36,1,1
%N A126465 Triangle T, read by rows, where row n equals row (n-1) of matrix power T^(n(n+1)/2) concatenated with a trailing '1', for n>0, with T(0,0) = 1.
%C A126465 Amazingly, A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1; and also A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457.
%e A126465 Triangle T begins:
%e A126465 1,
%e A126465 1, 1,
%e A126465 3, 1, 1,
%e A126465 33, 6, 1, 1,
%e A126465 855, 105, 10, 1, 1,
%e A126465 40475, 3710, 255, 15, 1, 1,
%e A126465 3039204, 219625, 11935, 525, 21, 1, 1,
%e A126465 331630320, 19545316, 879571, 31584, 966, 28, 1, 1,
%e A126465 49563943161, 2437990653, 93365328, 2856819, 72786, 1638, 36, 1, 1, ...
%e A126465 Matrix cube T^3 begins:
%e A126465 1;
%e A126465 [3, 1]; <-- row 1 of T^3 + '1' = row 2 of T;
%e A126465 12, 3, 1; ...
%e A126465 Matrix power T^6 begins:
%e A126465 1;
%e A126465 6, 1;
%e A126465 [33, 6, 1]; <-- row 2 of T^6 + '1' = row 3 of T.
%e A126465 Matrix power T^10 begins:
%e A126465 1;
%e A126465 10, 1;
%e A126465 75, 10, 1;
%e A126465 [855, 105, 10, 1]; <-- row 3 of T^10 + '1' = row 4 of T.
%e A126465 Matrix power T^15 begins:
%e A126465 1;
%e A126465 15, 1;
%e A126465 150, 15, 1;
%e A126465 1895, 195, 15, 1;
%e A126465 [40475, 3710, 255, 15, 1]; <-- row 4 of T^15 + '1' = row 5 of T.
%o A126465 (PARI) {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!,r-c))), N=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1,r-c)))); (N*M^-1)[n+1,k+1]}
%Y A126465 Columns: A126466, A126467, A126468; A126469 (row sums); A126460 (dual); A101479 (variant).
%K A126465 nonn,tabl
%O A126465 0,4
%A A126465 _Paul D. Hanna_, Dec 27 2006