cp's OEIS Frontend

A185620 Triangular matrix T that satisfies: T^3 - T^2 + I = SHIFT_LEFT(T), as read by rows.

%I A185620 #12 Mar 30 2012 18:37:25
%S A185620 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,
%T A185620 1525,1330,442,85,11,1,1,1,12555,12134,4345,897,126,13,1,1,1,123098,
%U A185620 129359,49114,10687,1586,175,15,1,1,1,1408656,1587501,632104,143335,22156,2557
%N A185620 Triangular matrix T that satisfies: T^3 - T^2 + I = SHIFT_LEFT(T), as read by rows.
%F A185620 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.
%F A185620 Let U equal T shifted up one diagonal; then U*T^2 equals U shifted left one column.
%e A185620 Triangle T begins:
%e A185620 1;
%e A185620 1, 1;
%e A185620 1, 1, 1;
%e A185620 1, 3, 1, 1;
%e A185620 1, 10, 5, 1, 1;
%e A185620 1, 42, 27, 7, 1, 1;
%e A185620 1, 226, 173, 52, 9, 1, 1;
%e A185620 1, 1525, 1330, 442, 85, 11, 1, 1;
%e A185620 1, 12555, 12134, 4345, 897, 126, 13, 1, 1;
%e A185620 1, 123098, 129359, 49114, 10687, 1586, 175, 15, 1, 1;
%e A185620 1, 1408656, 1587501, 632104, 143335, 22156, 2557, 232, 17, 1, 1;
%e A185620 1, 18499835, 22127494, 9167575, 2149761, 343091, 40936, 3858, 297, 19, 1, 1; ...
%e A185620 Matrix square T^2 begins:
%e A185620 1;
%e A185620 2, 1;
%e A185620 3, 2, 1;
%e A185620 6, 7, 2, 1;
%e A185620 18, 28, 11, 2, 1;
%e A185620 79, 142, 66, 15, 2, 1;
%e A185620 463, 913, 470, 120, 19, 2, 1;
%e A185620 3396, 7244, 3997, 1098, 190, 23, 2, 1;
%e A185620 ...
%e A185620 Matrix cube T^3 begins:
%e A185620 1;
%e A185620 3, 1;
%e A185620 6, 3, 1;
%e A185620 16, 12, 3, 1;
%e A185620 60, 55, 18, 3, 1;
%e A185620 305, 315, 118, 24, 3, 1;
%e A185620 1988, 2243, 912, 205, 30, 3, 1;
%e A185620 15951, 19378, 8342, 1995, 316, 36, 3, 1;
%e A185620 ...
%e A185620 Thus T^3 - T^2 + I begins:
%e A185620 1;
%e A185620 1, 1;
%e A185620 3, 1, 1;
%e A185620 10, 5, 1, 1;
%e A185620 42, 27, 7, 1, 1;
%e A185620 226, 173, 52, 9, 1, 1;
%e A185620 1525, 1330, 442, 85, 11, 1, 1;
%e A185620 12555, 12134, 4345, 897, 126, 13, 1, 1;
%e A185620 ...
%e A185620 which equals T shifted left one column.
%e A185620 ...
%e A185620 ALTERNATE GENERATING FORMULA.
%e A185620 Let U equal T shifted up one diagonal:
%e A185620 1;
%e A185620 1, 1;
%e A185620 1, 3, 1;
%e A185620 1, 10, 5, 1;
%e A185620 1, 42, 27, 7, 1;
%e A185620 1, 226, 173, 52, 9, 1;
%e A185620 1, 1525, 1330, 442, 85, 11, 1;
%e A185620 1, 12555, 12134, 4345, 897, 126, 13, 1;
%e A185620 ...
%e A185620 then U*T^2 begins:
%e A185620 1;
%e A185620 3, 1;
%e A185620 10, 5, 1;
%e A185620 42, 27, 7, 1;
%e A185620 226, 173, 52, 9, 1;
%e A185620 1525, 1330, 442, 85, 11, 1;
%e A185620 12555, 12134, 4345, 897, 126, 13, 1;
%e A185620 ...
%e A185620 which equals U shifted left one column.
%o A185620 (PARI) {T(n, k)=local(A=Mat(1), B); for(m=1, n, B=A^3-A^2+A^0;
%o A185620 A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2|j==i, A[i, j]=1,
%o A185620 if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return(A[n+1, k+1])}
%Y A185620 Cf. columns: A185621, A185622, A185623; A185624 (T^2), A185628 (T^3).
%Y A185620 Cf. variants: A104445, A185641.
%K A185620 nonn,tabl
%O A185620 0,8
%A A185620 _Paul D. Hanna_, Feb 01 2011