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.
1, 1, 1, 3, 1, 1, 33, 6, 1, 1, 855, 105, 10, 1, 1, 40475, 3710, 255, 15, 1, 1, 3039204, 219625, 11935, 525, 21, 1, 1, 331630320, 19545316, 879571, 31584, 966, 28, 1, 1, 49563943161, 2437990653, 93365328, 2856819, 72786, 1638, 36, 1, 1
Offset: 0
Examples
Triangle T begins: 1, 1, 1, 3, 1, 1, 33, 6, 1, 1, 855, 105, 10, 1, 1, 40475, 3710, 255, 15, 1, 1, 3039204, 219625, 11935, 525, 21, 1, 1, 331630320, 19545316, 879571, 31584, 966, 28, 1, 1, 49563943161, 2437990653, 93365328, 2856819, 72786, 1638, 36, 1, 1, ... Matrix cube T^3 begins: 1; [3, 1]; <-- row 1 of T^3 + '1' = row 2 of T; 12, 3, 1; ... Matrix power T^6 begins: 1; 6, 1; [33, 6, 1]; <-- row 2 of T^6 + '1' = row 3 of T. Matrix power T^10 begins: 1; 10, 1; 75, 10, 1; [855, 105, 10, 1]; <-- row 3 of T^10 + '1' = row 4 of T. Matrix power T^15 begins: 1; 15, 1; 150, 15, 1; 1895, 195, 15, 1; [40475, 3710, 255, 15, 1]; <-- row 4 of T^15 + '1' = row 5 of T.
Crossrefs
Programs
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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]}
Comments