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A343234
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Triangle T read by rows: lower triangular Riordan matrix of the Toeplitz type with first column A067687.
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2
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1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 12, 5, 2, 1, 1, 29, 12, 5, 2, 1, 1, 69, 29, 12, 5, 2, 1, 1, 165, 69, 29, 12, 5, 2, 1, 1, 393, 165, 69, 29, 12, 5, 2, 1, 1, 937, 393, 165, 69, 29, 12, 5, 2, 1, 1, 2233, 937, 393, 165, 69, 29, 12, 5, 2, 1, 1
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OFFSET
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0,4
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COMMENTS
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This infinite lower triangular Riordan matrix T is the so-called L-eigen-matrix of the infinite lower triangular Riordan matrix A027293 (but with offset 0 for rows and columns). Such eigentriangles have been considered by Paul Barry in the paper given as a link in A186020.
This means that E is the L-eigen-matrix of an infinite lower triangular matrix M if M*E = L*(E - I), with the unit matrix I and the matrix L with elements L(i, j) = delta_{i, j-1} (Kronecker's delta-symbol; first upper diagonal with 1's).
Therefore, this notion is analogous to calling sequence S an L-eigen-sequence of matrix M if M*vec(S) = L.vec(S) (or vec(S) is an eigensequence of M - L with eigenvalue 0), used by Bernstein and Sloane, see the links in A155002.
L*(E - I) is the E matrix after elimination of the main diagonal and then the first row, and starting with offset 0. Because for infinite lower triangular matrices L^{tr}.L = I (tr stands for transposed), this leads to M = L*(I - E^{-1}) or E = (I - L^{tr}*M)^{-1}.
Note that Gary W. Adamson uses a different notion: E is the eigentriangle of a triangle T if the columns of E are the columns j of T multiplied by the sequence elements B_j of B with o.g.f. x/(1 - x*G(x)), with the o.g.f. G(x) of column nr. 1 of T. Or E(i, j) = T(i, j)*B(j). In short: sequence B is the L-eigen-sequence of the infinite lower triangular matrix T (but with offset 1): T*vec(B) = L.vec(B). See, e.g., A143866.
Thanks to Gary W. Adamson for motivating my occupation with such eigentriangles and eigensequences.
The first column of the present triangle T is A067687, which is then shifted downwards (Riordan of Toeplitz type).
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LINKS
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FORMULA
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Matrix elements: T(n, m) = A067687(n-m), for n >= m >= 0, and 0 otherwise.
O.g.f. of row polynomials R(n,x) = Sum_{m=0..n} T(n, m)*x^m is
G(z, x) = 1/((1 - z*P(z))*(1 - x*z)), with the o.g.f. P of A000041 (number of partitions).
O.g.f. column m: G_m(x) = x^m/(1 - x*P(x)), for m >= 0.
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EXAMPLE
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The triangle T begins:
n \ m 0 1 2 3 4 5 6 7 8 9 ...
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0: 1
1: 1 1
2: 2 1 1
3: 5 2 1 1
4: 12 5 2 1 1
5: 29 12 5 2 1 1
6: 69 29 12 5 2 1 1
7: 165 69 29 12 5 2 1 1
8: 393 165 69 29 12 5 2 1 1
9: 937 393 165 69 29 12 5 2 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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