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A343233
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Triangle read by rows: Riordan triangle T = (1 - x*c(x), x), with the generating function c of A000108 (Catalan).
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1
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1, -1, 1, -1, -1, 1, -2, -1, -1, 1, -5, -2, -1, -1, 1, -14, -5, -2, -1, -1, 1, -42, -14, -5, -2, -1, -1, 1, -132, -42, -14, -5, -2, -1, -1, 1, -429, -132, -42, -14, -5, -2, -1, -1, 1, -1430, -429, -132, -42, -14, -5, -2, -1, -1, 1
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OFFSET
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0,7
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COMMENTS
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As an unsigned sequence a(n) this is identical with the one of A155586(n+1), for n >= 0, but the triangle is not a simple signed version of A155586. See the formula.
This lower triangular Riordan matrix T of Toeplitz type is the inverse of the Riordan matrix (c(x), x) = |A106270|, also of Toeplitz type.
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LINKS
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FORMULA
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The lower triangular matrix T satisfies: T = I - L^{tr}*|A106270|, also for the finite N X N version, with the unit matrix I and the lower triangular matrix L^{tr}(i, j) = delta_{i, j-1} (Kronecker symbol delta) with first lower diagonal of 1s and 0 otherwise.
T(n, n) = 1, and for T(n, m) = -C_{n - 1 - m } = - |A106270(n-1, m)|, for 0 <= m <= n-1, with the Catalan numbers C(n) = A000108, and T(n, m) = 0 for n < m.
O.g.f. of column m: (1/c(x)*x^m = (1 - x*c(x))*x^m (Riordan matrix of Toeplitz type), with the o.g.f. c of A000108.
O.g.f. row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m, that is the o.g.f. of the triangle. G(z, x) = c(z)/(1 - x*z).
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EXAMPLE
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The triangle matrix T begins:
n/m 0 1 2 3 4 5 6 7 8 9 ...
--------------------------------------------------
0: 1
1: -1 1
2: -1 -1 1
3: -2 -1 -1 1
4: -5 -2 -1 -1 1
5: -14 -5 -2 -1 -1 1
6: -42 -14 -5 -2 -1 -1 1
7: -132 -42 -14 -5 -2 -1 -1 1
8: -429 -132 -42 -14 -5 -2 -1 -1 1
9: -1430 -429 -132 -42 -14 -5 -2 -1 -1 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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