A374429 Triangle read by rows: T(n, k) = ((3*(-1)^k + 1)/2)*abs(qStirling2(n, k, -1)). Polynomials related to the Lucas and Fibonacci numbers.

2, 0, -1, 0, -1, 2, 0, -1, 2, -1, 0, -1, 2, -2, 2, 0, -1, 2, -3, 4, -1, 0, -1, 2, -4, 6, -3, 2, 0, -1, 2, -5, 8, -6, 6, -1, 0, -1, 2, -6, 10, -10, 12, -4, 2, 0, -1, 2, -7, 12, -15, 20, -10, 8, -1, 0, -1, 2, -8, 14, -21, 30, -20, 20, -5, 2

Offset: 0

Author

Peter Luschny, Jul 25 2024

Keywords

• sign
• tabl

Comments

The idea behind the Fibonacci and Lucas sequences is simple: Put the '2' in the middle of a band and place a '1' to the left and a '-1' to the right. Now add the sum of the two immediate numbers with higher indexes on the left side and on the right side the sum of the two with lower indexes. Schematically, after a few steps, it looks like this:

..., 18, 11, 7, 4, 3, 1, <-- + [2] + --> -1, 1, 0, 1, 1, 2, 3, 5, ...

This generates the Lucas sequence on the left (with a descending index) and the Fibonacci sequence on the right (with an ascending index). The fact that the first two terms (-1, 1) of the Fibonacci sequence were 'forgotten' in A000045 is from our point of view only a difference in the choice of the offset of the central term. Our choice is, in any case, consistent with Knuth's continuation of the Fibonacci numbers into the negative range (A039834).

This construction is to be captured in a family of polynomials. The idea is that the two sequences are the values of the polynomials at the points x = -1 and x = 1. This continues from A374439. Here we choose signed coefficients and shift the powers up by one.

This approach also reveals that another important sequence follows the same logic: the Pell numbers (A000129). These, and their dual counterpart A048654, are interpolated by the polynomials at the points x = 1/2 and x = -1/2 (up to the normalization factor 2^n). The table in the example section gives an overview.

The formal representation is based on the qStirling2 numbers in the version defined in SageMath (see also A065941 and A333143).

Examples

			Triangle starts:
  [0] [2]
  [1] [0, -1]
  [2] [0, -1, 2]
  [3] [0, -1, 2, -1]
  [4] [0, -1, 2, -2,  2]
  [5] [0, -1, 2, -3,  4,  -1]
  [6] [0, -1, 2, -4,  6,  -3,  2]
  [7] [0, -1, 2, -5,  8,  -6,  6,  -1]
  [8] [0, -1, 2, -6, 10, -10, 12,  -4, 2]
  [9] [0, -1, 2, -7, 12, -15, 20, -10, 8, -1]
.
Table of interpolated sequences:
  |    | 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*   |
  |  1 |        -1         |    1    |       1    |       1    |
  |  2 |         1         |    3    |       0    |       4    |
  |  3 |         0         |    4    |       1    |       9    |
  |  4 |         1         |    7    |       2    |      22    |
  |  5 |         1         |   11    |       5    |      53    |
  |  6 |         2         |   18    |      12    |     128    |
  |  7 |         3         |   29    |      29    |     309    |
  |  8 |         5         |   47    |      70    |     746    |
  |  9 |         8         |   76    |     169    |    1801    |
  | 10 |        13         |  123    |     408    |    4348    |
		

Crossrefs

Cf. A000045 (Fibonacci), A039834 (negaFibonacci), A000204 (Lucas), A000129 (Pell), A048654 (dual Pell), A065941 (qStirling2).

Cf. A374439 (variant).

Programs

SageMath

from sage.combinat.q_analogues import q_stirling_number2
def T(n, k):
    return ((3*(-1)^k + 1)//2)*abs(q_stirling_number2(n, k, -1))
for n in range(10): print([T(n, k) for k in range(n + 1)])
def P(n, x):
    if n < 0: return P(-n, -x)
    return sum(T(n, k)*x^k for k in range(n + 1))
# Lucas and Fibonacci combined
print([P(n, 1) for n in range(-6, 9)])
# Table of interpolated sequences
print("|    | A039834 & A000045 | A000032 |   A000129  |   A048654  |")
print("|  n |      P(n, 1)      | P(n,-1) |-2^nP(n,1/2)|2^nP(n,-1/2)|")
f = "| {0:2d} | {1:9d}         | {2:4d}    | {3:7d}    | {4:7d}    |"
for n in range(1, 11): print(f.format(n, P(n, 1), P(n, -1),
                       int(-2**n*P(n, 1/2)), int(2**n*P(n, -1/2))))