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
Examples
[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, ...
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.
Crossrefs
Programs
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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);
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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 *)
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