cp's OEIS Frontend

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).

%I A247506 #35 Aug 12 2022 12:38:10
%S A247506 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,
%T A247506 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,
%U A247506 1,0,1,1,2,4,8,16,31,56,81,55,1,0
%N 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).
%H A247506 Stefano Spezia, <a href="/A247506/b247506.txt">First 140 antidiagonals of the array, flattened</a>
%H A247506 Harold R. Parks and Dean C. Wills, <a href="https://arxiv.org/abs/2208.01224">Sum of k-bonacci Numbers</a>, arXiv:2208.01224 [math.CO], 2022. See p. 5.
%F A247506 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
%e A247506 [n\k] [0][1][2][3][4] [5] [6] [7]  [8]  [9] [10]  [11]  [12]
%e A247506    [0] 1, 0, 0, 0, 0,  0,  0,  0,   0,   0,   0,    0,    0
%e A247506    [1] 1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1,    1,    1
%e A247506    [2] 1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89,  144,  233  [A000045]
%e A247506    [3] 1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274,  504,  927  [A000073]
%e A247506    [4] 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401,  773, 1490  [A000078]
%e A247506    [5] 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464,  912, 1793  [A001591]
%e A247506    [6] 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492,  976, 1936  [A001592]
%e A247506    [7] 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000  [A066178]
%e A247506    [8] 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028  [A079262]
%e A247506    [.] .  .  .  .  .   .   .   .    .    .    .     .     .
%e A247506   [oo] 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048  [A011782]
%e A247506 .
%e A247506 As a triangular array, starts:
%e A247506   1,
%e A247506   1, 0,
%e A247506   1, 1, 0,
%e A247506   1, 1, 1, 0,
%e A247506   1, 1, 2, 1, 0,
%e A247506   1, 1, 2, 3, 1, 0,
%e A247506   1, 1, 2, 4, 5, 1, 0,
%e A247506   1, 1, 2, 4, 7, 8, 1, 0,
%e A247506   1, 1, 2, 4, 8, 13, 13, 1, 0,
%e A247506   1, 1, 2, 4, 8, 15, 24, 21, 1, 0,
%e A247506   ...
%p A247506 A := (n,k) -> coeff(series((1-add(x^j, j=1..n))^(-1),x,k+2),x,k):
%p A247506 seq(print(seq(A(n,k), k=0..12)), n=0..9);
%t A247506 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 *)
%Y A247506 Cf. A247505, A011782, A000045, A000073, A000078, A001591, A001592, A066178, A079262.
%Y A247506 Cf. A048887, A092921, A144406.
%K A247506 tabl,nonn
%O A247506 0,13
%A A247506 _Peter Luschny_, Nov 02 2014