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A244422
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Quasi-Riordan triangle ((2-z)/(1-z), -z^2/(1-z)). Row reversed monic Chebyshev T-polynomials without vanishing columns.
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4
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2, 1, 0, 1, -2, 0, 1, -3, 0, 0, 1, -4, 2, 0, 0, 1, -5, 5, 0, 0, 0, 1, -6, 9, -2, 0, 0, 0, 1, -7, 14, -7, 0, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, 0
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OFFSET
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0,1
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COMMENTS
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This is a signed version of the triangle A061896.
The coefficient table for the monic Chebyshev polynomials of the first kind R(n, x) = 2*T(n, x/2) is given in A127672. For the T-polynomials see A053120. The present table is obtained from the row reversed coefficient table A127672 by deleting all odd numbered columns which have only zeros, and appending in the rows numbered n >= 1 zeros in order to obtain a triangle. This becomes the quasi-Riordan triangle T = ((2-z)/(1-z), -z^2/(1-z)). This means that the o.g.f. of the row polynomials Rrev(n, x) := sqrt(x)^n*R(n, 1/sqrt(x))= sum(T(n, k)*x^k, k=0..n) have o.g.f. (2-z)/(1 - z + x*z^2) like for ordinary Riordan triangles. However this is not a Riordan triangle (or lower triangular infinite dimensional matrix) in the usual sense because it is not invertible. Therefore, this lower triangular matrix is not a member of the Riordan group.
The row sums give repeat(2,1,-1,-2,-1) which is A057079(n+1), n >= 0. The alternating row sums give the Lucas numbers A000032.
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LINKS
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FORMULA
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T(n,k) = [x^k] Rrev(n, x), k=0, 1, ..., n, with the row polynomials Rrev(n, x) = sqrt(x)^n*R(n,1/sqrt(x)), with R(n, x) given in A127672 (monic Chebyshev polynomials of the first kind).
O.g.f. row polynomials Rrev(n,x) = sum(T(n,k)*x^k, k=0..n): (2-z)/(1 - z + x*z^2) (quasi-Riordan).
Recurrence: T(n,k) = T(n-1, k) - T(n-2, k-1), n >= k >= 1, T(n,k) = 0 if n < k, T(0,0) = 2, T(n,0) = 1 if n>=1, (Compare with A061896).
For n >= 1 the entries without trailing zeros are given by T(n,k) = (-1)^k*(n/(n-k))*binomial(n-k,k),k=0..floor(n/2)).
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EXAMPLE
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The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11
0: 2
1: 1 0
2: 1 -2 0
3: 1 -3 0 0
4: 1 -4 2 0 0
5: 1 -5 5 0 0 0
6: 1 -6 9 -2 0 0 0
7: 1 -7 14 -7 0 0 0 0
8: 1 -8 20 -16 2 0 0 0 0
9: 1 -9 27 -30 9 0 0 0 0 0
10: 1 -10 35 -50 25 -2 0 0 0 0 0
11: 1 -11 44 -77 55 -11 0 0 0 0 0 0
...
Rrev(3, x) = 1 - 3*x = sqrt(x)^3*R(3,1/sqrt(x)) = sqrt(x)^3*(-3/sqrt(x) + 1/sqrt(x)^3 ) = -3*x + 1.
Rrev(4, x) = 1 - 4*x + 2*x^2 = sqrt(x)^4*(2 - 4/sqrt(x)^2 + 1/sqrt(x)^4) = 2*x^2 - 4*x + 1.
Recurrence: T(4,1) = T(3, 1) - T(2, 0) = -3 -1 = -4.
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CROSSREFS
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Cf. A053120, A127672, A054977, -A000027, A000096, -A005581, A005582, -A005583, A005584, A057079, A000032.
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KEYWORD
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AUTHOR
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STATUS
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approved
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