A352361 Array read by ascending antidiagonals. A(n, k) = Fibonacci(k, n), where Fibonacci(n, x) are the Fibonacci polynomials.

0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 0, 0, 1, 3, 5, 3, 1, 0, 1, 4, 10, 12, 5, 0, 0, 1, 5, 17, 33, 29, 8, 1, 0, 1, 6, 26, 72, 109, 70, 13, 0, 0, 1, 7, 37, 135, 305, 360, 169, 21, 1, 0, 1, 8, 50, 228, 701, 1292, 1189, 408, 34, 0, 0, 1, 9, 65, 357, 1405, 3640, 5473, 3927, 985, 55, 1

Offset: 0

Author

Peter Luschny, Mar 18 2022

Keywords

• nonn
• easy
• tabl

Comments

From Michael A. Allen, Mar 26 2023: (Start)

Row n is the n-metallonacci sequence for n>0.

A(n,k), for n > 0 and k > 0, is the number of tilings of a (k-1)-board (a board with dimensions (k-1) X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are n kinds of squares available. (End)

Examples

			Array, A(n,k), starts:
  n\k 0, 1, 2,  3,   4,    5,     6,      7,       8,        9, ...
  -------------------------------------------------------------------------
  [0] 0, 1, 0,  1,   0,    1,     0,      1,       0,        1, ... A000035;
  [1] 0, 1, 1,  2,   3,    5,     8,     13,      21,       34, ... A000045;
  [2] 0, 1, 2,  5,  12,   29,    70,    169,     408,      985, ... A000129;
  [3] 0, 1, 3, 10,  33,  109,   360,   1189,    3927,    12970, ... A006190;
  [4] 0, 1, 4, 17,  72,  305,  1292,   5473,   23184,    98209, ... A001076;
  [5] 0, 1, 5, 26, 135,  701,  3640,  18901,   98145,   509626, ... A052918;
  [6] 0, 1, 6, 37, 228, 1405,  8658,  53353,  328776,  2026009, ... A005668;
  [7] 0, 1, 7, 50, 357, 2549, 18200, 129949,  927843,  6624850, ... A054413;
  [8] 0, 1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, ... A041025;
  [9] 0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, ... A099371;
      |  |  |  | A054602 |   A124152;
      |  |  |  A002522   A057721;
      |  |  A001477;
      |  A000012;
      A000004;
Antidiagonals, T(n, k), begin as:
  0;
  0, 1;
  0, 1, 0;
  0, 1, 1,  1;
  0, 1, 2,  2,   0;
  0, 1, 3,  5,   3,   1;
  0, 1, 4, 10,  12,   5,   0;
  0, 1, 5, 17,  33,  29,   8,   1;
  0, 1, 6, 26,  72, 109,  70,  13,  0;
  0, 1, 7, 37, 135, 305, 360, 169, 21, 1;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened
• Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.

Crossrefs

Other versions of this array are A073133, A157103, A172236.

Rows n: A000035 (n=0), A000045 (n=1), A000129 (n=2), A006190 (n=3), A001076 (n=4), A052918 (n=5), A005668 (n=6), A054413 (n=7), A041025 (n=8), A099371 (n=9).

Columns k: A000004 (k=0), A000012 (k=1), A001477 (k=2), A002522 (k=3), A054602 (k=4), A057721 (k=5), A124152 (k=6).

Cf. A084844 (main diagonal), A352362 (Lucas polynomials), A350470 (Jacobsthal polynomials).

Sums include: A304357 (row sums), A304359.

Cf. A084845.

Programs

Magma

A352361:= func< n, k | k le 1 select k else Evaluate(DicksonSecond(k-1, -1), n-k) >;
[A352361(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Sep 29 2024

Maple

seq(seq(combinat:-fibonacci(k, n - k), k = 0..n), n = 0..11);

Mathematica

Table[Fibonacci[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten
(* or *)
A[n_, k_] := With[{s = Sqrt[n^2 + 4]}, ((n + s)^k - (n - s)^k) / (2^k*s)];
Table[Simplify[A[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm

PARI

A(n, k) = ([1, k; 1, k-1]^n)[2, 1] ;
export(A)
for(k = 0, 9, print(parvector(10, n, A(n - 1, k))))

SageMath

def A352361(n, k): return lucas_number1(k,n-k,-1)
flatten([[A352361(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 29 2024

Formula

A(n, k) = Sum_{j=0..floor((k-1)/2)} binomial(k-j-1, j)*n^(k-2*j-1).

A(n, k) = ((n + s)^k - (n - s)^k) / (2^k*s) where s = sqrt(n^2 + 4).

A(n, k) = [x^k] (x / (1 - n*x - x^2)).

A(n, k) = n^(k-1)*hypergeom([1 - k/2, 1/2 - k/2], [1 - k], -4/n^2) for n,k >= 1.

A(n, n) = T(2*n, n) = A084844(n).

From G. C. Greubel, Sep 29 2024: (Start)

T(n, k) = A(n-k, k) (antidiagonal triangle).

T(2*n+1, n+1) = A084845(n).

Sum_{k=0..n} T(n, k) = A304357(n) (row sums).

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)*A304359(n). (End)