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%I #11 May 09 2021 10:03:10
%S 1,1,1,2,1,2,3,2,2,5,5,3,4,5,12,8,5,6,10,12,29,13,8,10,15,24,29,70,21,
%T 13,16,25,36,58,70,169,34,21,26,40,60,87,140,169,408,55,34,42,65,96,
%U 145,210,338,408,985
%N Triangle read by rows, A104762 * (A000129 * 0^(n-k)).
%C Eigentriangle, row sums = rightmost term of next row.
%C Row sums = the Pell series starting with offset 1: (1, 2, 5, 12, 29, ...).
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%F Triangle read by rows, A104762 * (A000129 * 0^(n-k)).
%F A104762 = Fibonacci numbers "decrescendo", (1, 1, 2, 3, 5, ...) in every column.
%F (A000129 * 0^(n-k)) ) = the Pell series prefaced with a 1:
%F (1, 1, 2, 5, 12, ...) as the main diagonal and the rest zeros
%F From _Wolfdieter Lang_, Apr 13 2021: (Start)
%F T(n, m) = F(n+1-m)*A215928(m), with F = A000045, for n >= m >= 1, and 0 otherwise.
%F The lower triangular (infinite) matrix t with elements t(n, m) = T(n+1, m+1), for n >= m >= 0, and 0 otherwise, has row polynomials R(n, x) = Sum_{m=0..n} t(n, m)*x^m with o.g.f. G(z, x) = A(z)/(1 - x*z*A(x*z)) =
%F (1 - x*z - (x*z)^2)/((1 - z - z^2)*(1 - 2*x*z - (x*z)^2)), with the o.g.f. A(x) of (F_{n+1})_{n>=0}, where F = A000045.
%F The infinite dimensional lower triangular Riordan matrix TB := (1/(1 - x - x^2), x) (a Toeplitz matrix) with nonzero elements A104762(n+1, m+1) has sequence (A215928(m))_{m >=0} as 'L-eigen-sequence' (cf. the Bernstein-Sloane link for 'eigen-sequence'). This means that (TB - L)*vec(B) = 0-matrix, where L has elements L(i, j) = delta_{i, j-1} (first upper diagonal with 1s, otherwise 0), and the infinite vector vec(B) has the elements of A215928.
%F Thanks to _Gary W. Adamson_ for motivating me to look at such triangles and sequences. (End)
%e First ten rows of the triangle T(n, m):
%e n \ m 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 1
%e 2: 1 1
%e 3: 2 1 2
%e 4: 3 2 2 5
%e 5: 5 3 4 5 12
%e 6: 8 5 6 10 12 29
%e 7: 13 8 10 15 24 29 70
%e 8: 21 13 16 25 36 58 70 169
%e 9: 34 21 26 40 60 87 140 169 408
%e 10: 55 34 42 65 96 145 210 338 408 985
%e ... reformatted by - _Wolfdieter Lang_, Apr 13 2021
%e Row 4 = (3, 2, 2, 5) = termwise products of (3, 2, 1, 1) and (1, 1, 2, 5).
%Y Cf. A104762, A000045, A000129, A215928.
%K eigen,nonn,easy,tabl
%O 1,4
%A _Gary W. Adamson_ & _Roger L. Bagula_, Jan 18 2009