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.

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

0,3

Comments

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.

Links

Paolo Xausa, Rows n = 0..150 of the triangle, flattened

Peter Luschny, Illustrating the polynomials.

Formula

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

Example

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^nP(n,-1/2)|2^nP(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    |

Maple

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);

Mathematica

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 *)

Prog

(Python)
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)
(Python)
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
(Python)
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^nP(n, -1/2)|2^nP(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))))
(Magma)
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
(SageMath)
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

Crossrefs

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).

Sequence in context: A307322 A306737 A178474 * A164822 A361208 A110627

Adjacent sequences: A374436 A374437 A374438 * A374440 A374441 A374442

Keyword

nonn,tabl

Author

Peter Luschny, Jul 22 2024

Status

approved