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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.

%I A374439 #63 Mar 30 2025 04:25:33
%S A374439 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,
%T A374439 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,
%U A374439 35,40,15,10,1,1,2,10,18,36,56,56,70,35,30,6,2
%N 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.
%C A374439 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.
%C A374439 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.
%C A374439 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).
%C A374439 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.
%C A374439 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.
%H A374439 Paolo Xausa, <a href="/A374439/b374439.txt">Rows n = 0..150 of the triangle, flattened</a>
%H A374439 Peter Luschny, <a href="/A374439/a374439.pdf">Illustrating the polynomials</a>.
%F A374439 T(n, k) = 2^k' * binomial(n - k' - (k - k') / 2, (k - k') / 2) where k' = 1 if k is odd and otherwise 0.
%F A374439 T(n, k) = (1 + (k mod 2))*qStirling2(n, k, -1), see A333143.
%F A374439 2^n*P(n, -1/2) = A000129(n - 1), Pell numbers, P(-1) = 1.
%F A374439 2^n*P(n, 1/2) = A048654(n), dual Pell numbers.
%F A374439 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
%e A374439 Triangle starts:
%e A374439   [ 0] [1]
%e A374439   [ 1] [1, 2]
%e A374439   [ 2] [1, 2, 1]
%e A374439   [ 3] [1, 2, 2,  2]
%e A374439   [ 4] [1, 2, 3,  4,  1]
%e A374439   [ 5] [1, 2, 4,  6,  3,  2]
%e A374439   [ 6] [1, 2, 5,  8,  6,  6,  1]
%e A374439   [ 7] [1, 2, 6, 10, 10, 12,  4,  2]
%e A374439   [ 8] [1, 2, 7, 12, 15, 20, 10,  8,  1]
%e A374439   [ 9] [1, 2, 8, 14, 21, 30, 20, 20,  5,  2]
%e A374439   [10] [1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1]
%e A374439 .
%e A374439 Table of interpolated sequences:
%e A374439   |  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |
%e A374439   |  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|
%e A374439   |    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |
%e A374439   |  0 |        -1         |     1   |       1     |       1    |
%e A374439   |  1 |         1         |     3   |       0     |       4    |
%e A374439   |  2 |         0         |     4   |       1     |       9    |
%e A374439   |  3 |         1         |     7   |       2     |      22    |
%e A374439   |  4 |         1         |    11   |       5     |      53    |
%e A374439   |  5 |         2         |    18   |      12     |     128    |
%e A374439   |  6 |         3         |    29   |      29     |     309    |
%e A374439   |  7 |         5         |    47   |      70     |     746    |
%e A374439   |  8 |         8         |    76   |     169     |    1801    |
%e A374439   |  9 |        13         |   123   |     408     |    4348    |
%p A374439 A374439 := (n, k) -> ifelse(k::odd, 2, 1)*binomial(n - irem(k, 2) - iquo(k, 2), iquo(k, 2)):
%p A374439 # Alternative, using the function qStirling2 from A333143:
%p A374439 T := (n, k) -> 2^irem(k, 2)*qStirling2(n, k, -1):
%p A374439 seq(seq(T(n, k), k = 0..n), n = 0..10);
%t A374439 A374439[n_, k_] := (# + 1)*Binomial[n - (k + #)/2, (k - #)/2] & [Mod[k, 2]];
%t A374439 Table[A374439[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _Paolo Xausa_, Jul 24 2024 *)
%o A374439 (Python)
%o A374439 from functools import cache
%o A374439 @cache
%o A374439 def T(n: int, k: int) -> int:
%o A374439     if k > n: return 0
%o A374439     if k < 2: return k + 1
%o A374439     return T(n - 1, k) + T(n - 2, k - 2)
%o A374439 (Python)
%o A374439 from math import comb as binomial
%o A374439 def T(n: int, k: int) -> int:
%o A374439     o = k & 1
%o A374439     return binomial(n - o - (k - o) // 2, (k - o) // 2) << o
%o A374439 (Python)
%o A374439 def P(n, x):
%o A374439     if n < 0: return P(n, x)
%o A374439     return sum(T(n, k)*x**k for k in range(n + 1))
%o A374439 def sgn(x: int) -> int: return (x > 0) - (x < 0)
%o A374439 # Table of interpolated sequences
%o A374439 print("|  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |")
%o A374439 print("|  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|")
%o A374439 print("|    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |")
%o A374439 f = "| {0:2d} | {1:9d}         |  {2:4d}   |   {3:5d}     |    {4:4d}    |"
%o A374439 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))))
%o A374439 (Magma)
%o A374439 function T(n,k) // T = A374439
%o A374439   if k lt 0 or k gt n then return 0;
%o A374439   elif k le 1 then return k+1;
%o A374439   else return T(n-1,k) + T(n-2,k-2);
%o A374439   end if;
%o A374439 end function;
%o A374439 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 23 2025
%o A374439 (SageMath)
%o A374439 from sage.combinat.q_analogues import q_stirling_number2
%o A374439 def A374439(n,k): return (-1)^((k+1)//2)*2^(k%2)*q_stirling_number2(n+1, k+1, -1)
%o A374439 print(flatten([[A374439(n, k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 23 2025
%Y A374439 Triangles related to Lucas polynomials: A034807, A114525, A122075, A061896, A352362.
%Y A374439 Triangles related to Fibonacci polynomials: A162515, A053119, A168561, A049310, A374441.
%Y A374439 Sums include: A000204 (Lucas numbers, row), A000045 & A212804 (even sums, Fibonacci numbers), A006355 (odd sums), A039834 (alternating sign row).
%Y A374439 Type m^n*P(n, 1/m): A000129 & A048654 (Pell, m=2), A108300 & A003688 (m=3), A001077 & A048875 (m=4).
%Y A374439 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.
%Y A374439 Columns include: A040000 (k=1), A000027 (k=2), A005843 (k=3), A000217 (k=4), A002378 (k=5).
%Y A374439 Diagonals include: A000034 (k=n), A029578 (k=n-1), abs(A131259) (k=n-2).
%Y A374439 Cf. A029578 (subdiagonal), A124038 (row reversed triangle, signed).
%Y A374439 Cf. A039961, A065941 (qStirling2), A103631, A108299, A131259. A133607, A194005, A333143, A374438 (m=3).
%K A374439 nonn,tabl
%O A374439 0,3
%A A374439 _Peter Luschny_, Jul 22 2024