A215928 -id:A215928 - cp's OEIS frontend

A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 3, 1, 0, 5, 10, 9, 4, 1, 0, 8, 20, 22, 14, 5, 1, 0, 13, 38, 51, 40, 20, 6, 1, 0, 21, 71, 111, 105, 65, 27, 7, 1, 0, 34, 130, 233, 256, 190, 98, 35, 8, 1, 0, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 0, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1

Offset: 0

Philippe Deléham, Jan 21 2009

			Triangle begins:
[0] 1;
[1] 0,  1;
[2] 0,  1,   1;
[3] 0,  2,   2,   1;
[4] 0,  3,   5,   3,   1;
[5] 0,  5,  10,   9,   4,   1;
[6] 0,  8,  20,  22,  14,   5,  1;
[7] 0, 13,  38,  51,  40,  20,  6,  1;
[8] 0, 21,  71, 111, 105,  65, 27,  7, 1;
[9] 0, 34, 130, 233, 256, 190, 98, 35, 8, 1.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Row sums are in A215928.
Central terms: T(2*n,n) = A213684(n) for n > 0.
Cf. A000045, A037027, A122542, A059283, A321620.

a155161 n k = a155161_tabl !! n !! k
a155161_row n = a155161_tabl !! n
a155161_tabl = [1] : [0,1] : f [0] [0,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 17 2013

T := (n, k) -> binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..11); # Peter Luschny, May 23 2021
# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

CoefficientList[#, y]& /@ CoefficientList[(1-x-x^2)/(1-x-x^2-x*y)+O[x]^12, x] // Flatten (* Jean-François Alcover, Mar 01 2019 *)
(* Generates the triangle without the leading '1' (rows are rearranged). *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[x/(1 - x - x^2), 11] // Flatten  (* Peter Luschny, Feb 27 2021 *)

M(n,k):=pochhammer(n,k)/k!;
create_list(sum(M(k,i)*binomial(i,n-i-k),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 15 2011 */

T(n, k) given by [0,1,1,-1,0,0,0,...] DELTA [1,0,0,0,...] where DELTA is the operator defined in A084938.
a(n,k) = Sum_{i=0..n-k} M(k,i)*binomial(i,n-i-k), where M(n,k) = n(n+1)(n+2)...(n+k-1)/k!. - Emanuele Munarini, Mar 15 2011
Recurrence: a(n+2,k+1) = a(n+1,k+1) + a(n+1,k) + a(n,k+1). - Emanuele Munarini, Mar 15 2011
G.f.: (1-x-x^2)/(1-x-x^2-x*y). - Philippe Deléham, Feb 08 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n) (n > 0), A052991(n), A155179(n), A155181(n), A155195(n), A155196(n), A155197(n), A155198(n), A155199(n) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Feb 08 2012
T(n, k) = binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4). - Peter Luschny, May 23 2021

A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 3, 4, 1, 1, 2, 4, 6, 3, 2, 1, 2, 5, 8, 6, 6, 1, 1, 2, 6, 10, 10, 12, 4, 2, 1, 2, 7, 12, 15, 20, 10, 8, 1, 1, 2, 8, 14, 21, 30, 20, 20, 5, 2, 1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1, 1, 2, 10, 18, 36, 56, 56, 70, 35, 30, 6, 2

Offset: 0

Peter Luschny, Jul 22 2024

There are several versions of Lucas and Fibonacci polynomials in this database. Our naming follows the convention of calling polynomials after the values of the polynomials at x = 1. This assumes a regular sequence of polynomials, that is, a sequence of polynomials where degree(p(n)) = n. This view makes the coefficients of the polynomials (the terms of a row) a refinement of the values at the unity.
A remarkable property of the polynomials under consideration is that they are dual in this respect. This means they give the Lucas numbers at x = 1 and the Fibonacci numbers at x = -1 (except for the sign). See the example section.
The Pell numbers and the dual Pell numbers are also values of the polynomials, at the points x = -1/2 and x = 1/2 (up to the normalization factor 2^n). This suggests a harmonized terminology: To call 2^n*P(n, -1/2) = 1, 0, 1, 2, 5, ... the Pell numbers (A000129) and 2^n*P(n, 1/2) = 1, 4, 9, 22, ... the dual Pell numbers (A048654).
Based on our naming convention one could call A162515 (without the prepended 0) the Fibonacci polynomials. In the definition above only the initial values would change to: T(n, k) = k + 1 for k < 1. To extend this line of thought we introduce A374438 as the third triangle of this family.
The triangle is closely related to the qStirling2 numbers at q = -1. For the definition of these numbers see A333143. This relates the triangle to A065941 and A103631.

			Triangle starts:
  [ 0] [1]
  [ 1] [1, 2]
  [ 2] [1, 2, 1]
  [ 3] [1, 2, 2,  2]
  [ 4] [1, 2, 3,  4,  1]
  [ 5] [1, 2, 4,  6,  3,  2]
  [ 6] [1, 2, 5,  8,  6,  6,  1]
  [ 7] [1, 2, 6, 10, 10, 12,  4,  2]
  [ 8] [1, 2, 7, 12, 15, 20, 10,  8,  1]
  [ 9] [1, 2, 8, 14, 21, 30, 20, 20,  5,  2]
  [10] [1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1]
.
Table of interpolated sequences:
  |  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |
  |  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|
  |    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |
  |  0 |        -1         |     1   |       1     |       1    |
  |  1 |         1         |     3   |       0     |       4    |
  |  2 |         0         |     4   |       1     |       9    |
  |  3 |         1         |     7   |       2     |      22    |
  |  4 |         1         |    11   |       5     |      53    |
  |  5 |         2         |    18   |      12     |     128    |
  |  6 |         3         |    29   |      29     |     309    |
  |  7 |         5         |    47   |      70     |     746    |
  |  8 |         8         |    76   |     169     |    1801    |
  |  9 |        13         |   123   |     408     |    4348    |

Paolo Xausa, Rows n = 0..150 of the triangle, flattened
Peter Luschny, Illustrating the polynomials.

Triangles related to Lucas polynomials: A034807, A114525, A122075, A061896, A352362.
Triangles related to Fibonacci polynomials: A162515, A053119, A168561, A049310, A374441.
Sums include: A000204 (Lucas numbers, row), A000045 & A212804 (even sums, Fibonacci numbers), A006355 (odd sums), A039834 (alternating sign row).
Type m^n*P(n, 1/m): A000129 & A048654 (Pell, m=2), A108300 & A003688 (m=3), A001077 & A048875 (m=4).
Adding and subtracting the values in a row of the table (plus halving the values obtained in this way): A022087, A055389, A118658, A052542, A163271, A371596, A324969, A212804, A077985, A069306, A215928.
Columns include: A040000 (k=1), A000027 (k=2), A005843 (k=3), A000217 (k=4), A002378 (k=5).
Diagonals include: A000034 (k=n), A029578 (k=n-1), abs(A131259) (k=n-2).
Cf. A029578 (subdiagonal), A124038 (row reversed triangle, signed).
Cf. A039961, A065941 (qStirling2), A103631, A108299, A131259. A133607, A194005, A333143, A374438 (m=3).

function T(n,k) // T = A374439
  if k lt 0 or k gt n then return 0;
  elif k le 1 then return k+1;
  else return T(n-1,k) + T(n-2,k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2025

A374439 := (n, k) -> ifelse(k::odd, 2, 1)*binomial(n - irem(k, 2) - iquo(k, 2), iquo(k, 2)):
# Alternative, using the function qStirling2 from A333143:
T := (n, k) -> 2^irem(k, 2)*qStirling2(n, k, -1):
seq(seq(T(n, k), k = 0..n), n = 0..10);

A374439[n_, k_] := (# + 1)*Binomial[n - (k + #)/2, (k - #)/2] & [Mod[k, 2]];
Table[A374439[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, Jul 24 2024 *)

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k > n: return 0
    if k < 2: return k + 1
    return T(n - 1, k) + T(n - 2, k - 2)

from math import comb as binomial
def T(n: int, k: int) -> int:
    o = k & 1
    return binomial(n - o - (k - o) // 2, (k - o) // 2) << o

def P(n, x):
    if n < 0: return P(n, x)
    return sum(T(n, k)*x**k for k in range(n + 1))
def sgn(x: int) -> int: return (x > 0) - (x < 0)
# Table of interpolated sequences
print("|  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |")
print("|  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|")
print("|    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |")
f = "| {0:2d} | {1:9d}         |  {2:4d}   |   {3:5d}     |    {4:4d}    |"
for n in range(10): print(f.format(n, -P(n, -1), P(n, 1), int(2**n*P(n, -1/2)), int(2**n*P(n, 1/2))))

from sage.combinat.q_analogues import q_stirling_number2
def A374439(n,k): return (-1)^((k+1)//2)*2^(k%2)*q_stirling_number2(n+1, k+1, -1)
print(flatten([[A374439(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 23 2025

T(n, k) = 2^k' * binomial(n - k' - (k - k') / 2, (k - k') / 2) where k' = 1 if k is odd and otherwise 0.
T(n, k) = (1 + (k mod 2))*qStirling2(n, k, -1), see A333143.
2^n*P(n, -1/2) = A000129(n - 1), Pell numbers, P(-1) = 1.
2^n*P(n, 1/2) = A048654(n), dual Pell numbers.
T(2*n, n) = (1/2)*(-1)^n*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)*A045721((n-1)/2) ). - G. C. Greubel, Jan 23 2025

A105070 T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 8, 5, 20, 4, 6, 40, 24, 7, 70, 84, 8, 8, 112, 224, 64, 9, 168, 504, 288, 16, 10, 240, 1008, 960, 160, 11, 330, 1848, 2640, 880, 32, 12, 440, 3168, 6336, 3520, 384, 13, 572, 5148, 13728, 11440, 2496, 64, 14, 728, 8008, 27456, 32032, 11648, 896, 15, 910, 12012, 51480, 80080, 43680, 6720, 128

Offset: 1

Emeric Deutsch, Apr 05 2005

Row n contains ceiling(n/2) terms. Row sums yield the Pell numbers (A000129). Column 1 yields A007290.
Eigenvector equals A118397, so that A118397(n) = Sum_{k=0..[n/2]} T(n+1,k)*A118397(k) for n >= 0. - Paul D. Hanna, May 08 2006
Essentially a triangle, read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 07 2012

			Triangle begins:
  1;
  2;
  3,  2;
  4,  8;
  5, 20,  4;
  6, 40, 24;
(2, -1/2, 1/2, 0, 0, ...) DELTA (0, 1, -1, 0, 0, ...) begins:
  1;
  2,  0;
  3,  2,  0;
  4,  8,  0,  0;
  5, 20,  4,  0,  0;
  6, 40, 24,  0,  0,  0.
(1, 1, -1, 1, 0, 0, ...) DELTA (0, 0, 2, -2, 0, 0, ...) begins:
  1;
  1,  0;
  2,  0,  0;
  3,  2,  0,  0;
  4,  8,  0,  0,  0;
  5, 20,  4,  0,  0,  0;
  6, 40, 24,  0,  0,  0,  0. - _Philippe Deléham_, Apr 07 2012

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.
J. Ivie, Problem B-161, Fibonacci Quarterly, 8 (1970), 107-108.

Cf. A000129, A007290.

Cf. A118397 (eigenvector).

[2^k*Binomial(n,2*k+1): k in [0..Floor((n-1)/2)], n in [1..15]]; // G. C. Greubel, Mar 15 2020

T:=(n,k)->binomial(n,2*k+1)*2^k:for n from 1 to 15 do seq(T(n,k),k=0..floor((n-1)/2)) od; # yields sequence in triangular form

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207536 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A105070 *)
(* Clark Kimberling, Feb 18 2010 *)
Table[2^k*Binomial[n, 2*k+1], {n, 15}, {k,0,Floor[(n-1)/2]}]//Flatten (* G. C. Greubel, Mar 15 2020 *)

[[2^k*binomial(n,2*k+1) for k in (0..floor((n-1)/2))] for n in (1..15)] # G. C. Greubel, Mar 15 2020

E.g.f.: exp(x)*sinh(x*sqrt(2*y))/sqrt(2*y), cf. A034867. - Vladeta Jovovic, Apr 06 2005
From Philippe Deléham, Apr 07 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x+x^2-y*x^2)/(1-2*x+x^2-2*y*x^2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(1,1) = T(2,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k<0 or if k>n. (End)
Sum_{k=0..floor((n-1)/2)} T(n,k) = { P(n) (A000129(n)), A215928(n), (-1)^(n-1) *A077985(n-1), -A176981(n+1), (-1)^(n-1)*A215936(n+2) }, for n >= 1. - G. C. Greubel, Mar 15 2020

A215936 a(n) = -2*a(n-1) + a(n-2) for n > 2, with a(0) = a(1) = 1, a(2) = 0.

Original entry on oeis.org

1, 1, 0, 1, -2, 5, -12, 29, -70, 169, -408, 985, -2378, 5741, -13860, 33461, -80782, 195025, -470832, 1136689, -2744210, 6625109, -15994428, 38613965, -93222358, 225058681, -543339720, 1311738121, -3166815962, 7645370045, -18457556052, 44560482149

Offset: 0

Michael Somos, Aug 28 2012

BINOMIAL transform is A052955.

Essentially the same as A000129, A069306, A048624, A215928, A077985, and A176981. - R. J. Mathar, Sep 08 2013

			G.f. = 1 + x + x^3 - 2*x^4 + 5*x^5 - 12*x^6 + 29*x^7 - 70*x^8 + 169*x^9 - 408*x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
Index entries for linear recurrences with constant coefficients, signature (-2,1).

Cf. A001542, A001653, A052955, A077985.

[1,1] cat [n le 2 select (n-1) else -2*Self(n-1)+Self(n-2): n in [1..35] ]; // Vincenzo Librandi, Sep 09 2013

CoefficientList[Series[(1 + 3 x + x^2)/(1 + 2 x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 09 2013 *)
a[ n_] := With[ {m = If[ n < 1, 1 - n, n], s = If[ n < 1, (-1)^n, 1]}, s SeriesCoefficient[ x (1 + 2 x) / (1 + 2 x - x^2), {x, 0, m}]]; (* Michael Somos, Mar 19 2019 *)

{a(n) = my(m=n, s=1); if(n<1, m=1-n; s=(-1)^n); s * polcoeff( x * (1 + 2*x) / (1 + 2*x - x^2) + x * O(x^m), m)}; /* Michael Somos, Mar 19 2019 */

G.f.: 1 / (1 - x / (1 + x / (1 + x / (1 + x)))) = (1 + 3*x + x^2) / (1 + 2*x - x^2).
a(n + 3) = A077985(n). a(n) * a(n+2) - a(n+1)^2 = -(-1)^n.
a(2*n + 1) = A001653(n). a(2*n + 2) = -A001542(n).
a(n) = Sum_{k=0..n} A147746(n,k)*(-1)^(n-k). - Philippe Deléham, Aug 30 2012
G.f.: 1 + x + x^2/(1-x)  - G(0)*x^2 /(2-2*x), where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 10 2013
a(n) = (-1)^n a(1-n) = A000129(-1-n) if n < 0. a(n-2) = 2*a(n-1) + a(n) if n<1 or n>2. - Michael Somos, Mar 19 2019
E.g.f.: exp(-x)*(4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/2 - 1. - Stefano Spezia, Oct 31 2024

A055519 a(n) = 9a(n-1) + 33a(n-2) - 76a(n-3) - 33a(n-4) + 9*a(n-5) + a(n-6), a(0)=a(1)=1, a(2)=2, a(3)=35, a(4)=312, a(5)=3779.

Original entry on oeis.org

1, 1, 2, 35, 312, 3779, 41590, 474169, 5342808, 60450145, 682988978, 7720432691, 87256315920, 986227664411, 11146765278382, 125986353493225, 1423957841588232, 16094263592763889, 181905138292910570, 2055979904686591259, 23237679087969620328, 262643489044489470155

Offset: 0

Barry Cipra, Jul 04 2000

Index entries for linear recurrences with constant coefficients, signature (9,33,-76,-33,9,1).

Cf. A000045, A054894, A055518, A056572, A215928.

LinearRecurrence[{9,33,-76,-33,9,1},{1,1,2,35,312,3779},20] (* Harvey P. Dale, Oct 20 2021 *)

a(n) = Sum_{k=1..n} Fibonacci(k)^5*a(n-k), a(0)=1. - Vladeta Jovovic, Apr 23 2003

G.f.: (x^2+x-1)*(x^2+11*x-1)*(x^2-4*x-1)/(x^6+9*x^5-33*x^4-76*x^3+33*x^2+9*x-1). - Alois P. Heinz, Oct 24 2021

a(0)=1 prepended and edited by Alois P. Heinz, Oct 24 2021

A155002 Triangle read by rows, A104762 * (A000129 * 0^(n-k)).

Original entry on oeis.org

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, 13, 16, 25, 36, 58, 70, 169, 34, 21, 26, 40, 60, 87, 140, 169, 408, 55, 34, 42, 65, 96, 145, 210, 338, 408, 985

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jan 18 2009

Eigentriangle, row sums = rightmost term of next row.

Row sums = the Pell series starting with offset 1: (1, 2, 5, 12, 29, ...).

			First ten rows of the triangle T(n, m):
n \ m   1  2  3  4  5   6   7   8   9  10 ...
1:      1
2:      1  1
3:      2  1  2
4:      3  2  2  5
5:      5  3  4  5 12
6:      8  5  6 10 12  29
7:     13  8 10 15 24  29  70
8:     21 13 16 25 36  58  70 169
9:     34 21 26 40 60  87 140 169 408
10:    55 34 42 65 96 145 210 338 408 985
... reformatted by - _Wolfdieter Lang_, Apr 13 2021
Row 4 = (3, 2, 2, 5) = termwise products of (3, 2, 1, 1) and (1, 1, 2, 5).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A104762, A000045, A000129, A215928.

Triangle read by rows, A104762 * (A000129 * 0^(n-k)).
A104762 = Fibonacci numbers "decrescendo", (1, 1, 2, 3, 5, ...) in every column.
(A000129 * 0^(n-k)) ) = the Pell series prefaced with a 1:
(1, 1, 2, 5, 12, ...) as the main diagonal and the rest zeros
From Wolfdieter Lang, Apr 13 2021: (Start)
T(n, m) = F(n+1-m)*A215928(m), with F = A000045, for n >= m >= 1, and 0 otherwise.
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)) =
(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.
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.
Thanks to Gary W. Adamson for motivating me to look at such triangles and sequences. (End)

cp's OEIS Frontend

A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.

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A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

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A105070 T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.

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A215936 a(n) = -2*a(n-1) + a(n-2) for n > 2, with a(0) = a(1) = 1, a(2) = 0.

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A055519 a(n) = 9*a(n-1) + 33*a(n-2) - 76*a(n-3) - 33*a(n-4) + 9*a(n-5) + a(n-6), a(0)=a(1)=1, a(2)=2, a(3)=35, a(4)=312, a(5)=3779.

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A155002 Triangle read by rows, A104762 * (A000129 * 0^(n-k)).

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A308940 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} Fibonacci(k)*x^k/k!).

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A055519 a(n) = 9a(n-1) + 33a(n-2) - 76a(n-3) - 33a(n-4) + 9*a(n-5) + a(n-6), a(0)=a(1)=1, a(2)=2, a(3)=35, a(4)=312, a(5)=3779.