A247506 Generalized Fibonacci numbers: square array A(n,k) read by ascending antidiagonals, A(n,k) = [x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0, k>=0).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 5, 1, 0, 1, 1, 2, 4, 7, 8, 1, 0, 1, 1, 2, 4, 8, 13, 13, 1, 0, 1, 1, 2, 4, 8, 15, 24, 21, 1, 0, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 0, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 0

Offset

0,13

Links

Stefano Spezia, First 140 antidiagonals of the array, flattened

Harold R. Parks and Dean C. Wills, Sum of k-bonacci Numbers, arXiv:2208.01224 [math.CO], 2022. See p. 5.

Formula

A(n, k) = Sum_{j=0..floor(k/(n+1))} (-1)^j*((k - j*n) + j + delta(k,0))/(2*(k - j*n) + delta(k,0))*binomial(k - j*n, j)*2^(k-j*(n+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - Stefano Spezia, Aug 06 2022

Example

[n\k] [0][1][2][3][4] [5] [6] [7]  [8]  [9] [10]  [11]  [12]
   [0] 1, 0, 0, 0, 0,  0,  0,  0,   0,   0,   0,    0,    0
   [1] 1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1,    1,    1
   [2] 1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89,  144,  233  [A000045]
   [3] 1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274,  504,  927  [A000073]
   [4] 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401,  773, 1490  [A000078]
   [5] 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464,  912, 1793  [A001591]
   [6] 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492,  976, 1936  [A001592]
   [7] 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000  [A066178]
   [8] 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028  [A079262]
   [.] .  .  .  .  .   .   .   .    .    .    .     .     .
  [oo] 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048  [A011782]
.
As a triangular array, starts:
  1,
  1, 0,
  1, 1, 0,
  1, 1, 1, 0,
  1, 1, 2, 1, 0,
  1, 1, 2, 3, 1, 0,
  1, 1, 2, 4, 5, 1, 0,
  1, 1, 2, 4, 7, 8, 1, 0,
  1, 1, 2, 4, 8, 13, 13, 1, 0,
  1, 1, 2, 4, 8, 15, 24, 21, 1, 0,
  ...

Maple

A := (n, k) -> coeff(series((1-add(x^j, j=1..n))^(-1), x, k+2), x, k):
seq(print(seq(A(n, k), k=0..12)), n=0..9);

Mathematica

A[n_, k_] := A[n, k] = If[k<0, 0, If[k==0, 1, Sum[A[n, j], {j, k-n, k-1}]]]; Table[A[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 08 2019 *)

Crossrefs

Cf. A247505, A011782, A000045, A000073, A000078, A001591, A001592, A066178, A079262.

Cf. A048887, A092921, A144406.

Sequence in context: A290217 A293285 A262553 * A182172 A143841 A276719

Adjacent sequences: A247503 A247504 A247505 * A247507 A247508 A247509

Keyword

tabl,nonn

Author

Peter Luschny, Nov 02 2014

Status

approved