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