This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343234 #32 Apr 08 2025 08:45:45
%S A343234 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,
%T A343234 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,
%U A343234 2233,937,393,165,69,29,12,5,2,1,1
%N A343234 Triangle T read by rows: lower triangular Riordan matrix of the Toeplitz type with first column A067687.
%C A343234 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.
%C A343234 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).
%C A343234 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.
%C A343234 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}.
%C A343234 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 no. 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.
%C A343234 Thanks to _Gary W. Adamson_ for motivating my occupation with such eigentriangles and eigensequences.
%C A343234 The first column of the present triangle T is A067687, which is then shifted downwards (Riordan of Toeplitz type).
%F A343234 Matrix elements: T(n, m) = A067687(n-m), for n >= m >= 0, and 0 otherwise.
%F A343234 O.g.f. of row polynomials R(n,x) = Sum_{m=0..n} T(n, m)*x^m is
%F A343234 G(z, x) = 1/((1 - z*P(z))*(1 - x*z)), with the o.g.f. P of A000041 (number of partitions).
%F A343234 O.g.f. column m: G_m(x) = x^m/(1 - x*P(x)), for m >= 0.
%e A343234 The triangle T begins:
%e A343234 n \ m 0 1 2 3 4 5 6 7 8 9 ...
%e A343234 -----------------------------------------
%e A343234 0: 1
%e A343234 1: 1 1
%e A343234 2: 2 1 1
%e A343234 3: 5 2 1 1
%e A343234 4: 12 5 2 1 1
%e A343234 5: 29 12 5 2 1 1
%e A343234 6: 69 29 12 5 2 1 1
%e A343234 7: 165 69 29 12 5 2 1 1
%e A343234 8: 393 165 69 29 12 5 2 1 1
%e A343234 9: 937 393 165 69 29 12 5 2 1 1
%e A343234 ...
%Y A343234 Cf. A000041, A027293, A067687, A143866, A155002, A186020.
%K A343234 nonn,easy,tabl
%O A343234 0,4
%A A343234 _Wolfdieter Lang_, Apr 16 2021