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A140071
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Triangle read by rows: iterates of X * [1,0,0,0,...]; where X = an infinite lower bidiagonal matrix with [3,1,3,1,3,1...] in the main diagonal and [1,1,1,...] in the subdiagonal.
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3
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1, 3, 1, 9, 4, 1, 27, 13, 7, 1, 81, 40, 34, 8, 1, 243, 121, 142, 42, 11, 1, 729, 364, 547, 184, 75, 12, 1, 2187, 1093, 2005, 731, 409, 87, 15, 1, 6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1, 19683, 9841, 24604, 9844, 8610, 2454, 892, 148, 19, 1
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OFFSET
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1,2
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COMMENTS
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Companion triangle A140070 uses an analogous operation with the main diagonal [1,3,1,3,1,3,...].
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LINKS
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FORMULA
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O.g.f. (1 + (x - 1)*z)/(1 - 4*z - (x^2 - 3)*z^2) = 1 + (x + 3)*z + (x^2 + 4*x + 9)*z^2 + ....
Recurrence equation: T(n,k) = 4*T(n-1,k) - 3*T(n-2,k) + T(n-2,k-2).
Recurrence equation for row polynomials: R(n,x) = 4*R(n-1,x) + (x^2 - 3)*R(n-2,x) with R(0,x) = 1 and R(1,x) = 3 + x.
Another recurrence equation: R(n,x) = (x + 2)*R(n-1,x) + R(n-1,-x) with R(0,x) = 1. Cf. A157751. (End)
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EXAMPLE
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First few rows of the triangle are:
1;
3, 1;
9, 4, 1;
27, 13, 7, 1;
81, 40, 34, 8, 1;
243, 121, 142, 42, 11, 1;
729, 364, 547, 184, 75, 12, 1;
2187, 1093, 2005, 731, 409, 87, 15, 1;
6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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