OFFSET
0,4
COMMENTS
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).
FORMULA
EXAMPLE
The triangle T begins:
n \ m 0 1 2 3 4 5 6 7 8 9 ...
-----------------------------------------
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
...
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 16 2021
STATUS
approved