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 A185620 Triangular matrix T that satisfies: T^3 - T^2 + I = SHIFT_LEFT(T), as read by rows. 11
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Table of n, a(n) for n=0..61. 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. EXAMPLE 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. PROG (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])} CROSSREFS Cf. columns: A185621, A185622, A185623; A185624 (T^2), A185628 (T^3). Cf. variants: A104445, A185641. Sequence in context: A307847 A080214 A263383 * A096066 A294746 A326180 Adjacent sequences: A185617 A185618 A185619 * A185621 A185622 A185623 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Feb 01 2011 STATUS approved

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Last modified June 6 18:49 EDT 2023. Contains 363150 sequences. (Running on oeis4.)