A185620 Triangular matrix T that satisfies: T^3 - T^2 + I = SHIFT_LEFT(T), as read by rows.
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 10, 5, 1, 1, 1, 42, 27, 7, 1, 1, 1, 226, 173, 52, 9, 1, 1, 1, 1525, 1330, 442, 85, 11, 1, 1, 1, 12555, 12134, 4345, 897, 126, 13, 1, 1, 1, 123098, 129359, 49114, 10687, 1586, 175, 15, 1, 1, 1, 1408656, 1587501, 632104, 143335, 22156, 2557
Offset: 0
Examples
Triangle T begins: 1; 1, 1; 1, 1, 1; 1, 3, 1, 1; 1, 10, 5, 1, 1; 1, 42, 27, 7, 1, 1; 1, 226, 173, 52, 9, 1, 1; 1, 1525, 1330, 442, 85, 11, 1, 1; 1, 12555, 12134, 4345, 897, 126, 13, 1, 1; 1, 123098, 129359, 49114, 10687, 1586, 175, 15, 1, 1; 1, 1408656, 1587501, 632104, 143335, 22156, 2557, 232, 17, 1, 1; 1, 18499835, 22127494, 9167575, 2149761, 343091, 40936, 3858, 297, 19, 1, 1; ... Matrix square T^2 begins: 1; 2, 1; 3, 2, 1; 6, 7, 2, 1; 18, 28, 11, 2, 1; 79, 142, 66, 15, 2, 1; 463, 913, 470, 120, 19, 2, 1; 3396, 7244, 3997, 1098, 190, 23, 2, 1; ... Matrix cube T^3 begins: 1; 3, 1; 6, 3, 1; 16, 12, 3, 1; 60, 55, 18, 3, 1; 305, 315, 118, 24, 3, 1; 1988, 2243, 912, 205, 30, 3, 1; 15951, 19378, 8342, 1995, 316, 36, 3, 1; ... Thus T^3 - T^2 + I begins: 1; 1, 1; 3, 1, 1; 10, 5, 1, 1; 42, 27, 7, 1, 1; 226, 173, 52, 9, 1, 1; 1525, 1330, 442, 85, 11, 1, 1; 12555, 12134, 4345, 897, 126, 13, 1, 1; ... which equals T shifted left one column. ... ALTERNATE GENERATING FORMULA. Let U equal T shifted up one diagonal: 1; 1, 1; 1, 3, 1; 1, 10, 5, 1; 1, 42, 27, 7, 1; 1, 226, 173, 52, 9, 1; 1, 1525, 1330, 442, 85, 11, 1; 1, 12555, 12134, 4345, 897, 126, 13, 1; ... then U*T^2 begins: 1; 3, 1; 10, 5, 1; 42, 27, 7, 1; 226, 173, 52, 9, 1; 1525, 1330, 442, 85, 11, 1; 12555, 12134, 4345, 897, 126, 13, 1; ... which equals U shifted left one column.
Crossrefs
Programs
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PARI
{T(n, k)=local(A=Mat(1), B); for(m=1, n, B=A^3-A^2+A^0; A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2|j==i, A[i, j]=1, if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return(A[n+1, k+1])}
Formula
Recurrence: T(n+1,k+1) = [T^3](n,k) - [T^2](n,k) + [T^0](n,k) for n>=k>=0, with T(n,0)=1 for n>=0.
Let U equal T shifted up one diagonal; then U*T^2 equals U shifted left one column.