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%I #16 Feb 25 2020 16:30:38
%S 1,1,0,1,1,0,1,1,1,0,1,1,2,1,0,1,1,3,3,1,0,1,1,4,4,5,1,0,1,1,5,5,9,8,
%T 1,0,1,1,6,6,14,14,13,1,0,1,1,7,7,20,20,28,21,1,0,1,1,8,8,27,27,48,47,
%U 34,1,0,1,1,9,9,35,35,75,75,89,55,1,0
%N Generalized Fibonacci numbers. Square array read by ascending antidiagonals. F(n,k) for n >= 0 and k >= 0.
%H Genki Shibukawa, <a href="https://arxiv.org/abs/1907.00334">New identities for some symmetric polynomials and their applications</a>, arXiv:1907.00334 [math.CA], 2019.
%F F(n, k) = Sum_{j=0..(n-1)/2} (-1)^j*binomial(n-1-j,j)*F(n, k-1-2*j) + Sum_{j=0..(n-2)/2} (-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for k > 0; F(n, 0) = 1 and F(n, k) = 0 if k < 0.
%e Array starts:
%e [0] 1, 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, ...
%e [2] 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
%e [3] 1, 1, 3, 4, 9, 14, 28, 47, 89, 155, 286, 507, ...
%e [4] 1, 1, 4, 5, 14, 20, 48, 75, 165, 274, 571, 988, ...
%e [5] 1, 1, 5, 6, 20, 27, 75, 110, 275, 429, 1001, 1637, ...
%e [6] 1, 1, 6, 7, 27, 35, 110, 154, 429, 637, 1638, 2548, ...
%e [7] 1, 1, 7, 8, 35, 44, 154, 208, 637, 910, 2548, 3808, ...
%e [8] 1, 1, 8, 9, 44, 54, 208, 273, 910, 1260, 3808, 5508, ...
%e [9] 1, 1, 9, 10, 54, 65, 273, 350, 1260, 1700, 5508, 7752, ...
%o (SageMath)
%o @cached_function
%o def F(n, k):
%o if k < 0: return 0
%o if k == 0: return 1
%o a = sum((-1)^j*binomial(n-1-j,j )*F(n,k-1-2*j) for j in (0..(n-1)/2))
%o b = sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
%o return a + b
%o print([F(n-k, k) for n in (0..11) for k in (0..n)])
%Y Cf. A000007 (n=0), A000012 (n=1), A000045 (n=2), A006053 (n=3), A188021 (n=4), A231181 (n=5).
%K nonn,tabl
%O 0,13
%A _Peter Luschny_, Aug 21 2019
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