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A309896 Generalized Fibonacci numbers. Square array read by ascending antidiagonals. F(n,k) for n >= 0 and k >= 0.

%I A309896 #16 Feb 25 2020 16:30:38
%S A309896 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 A309896 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 A309896 34,1,0,1,1,9,9,35,35,75,75,89,55,1,0
%N A309896 Generalized Fibonacci numbers. Square array read by ascending antidiagonals. F(n,k) for n >= 0 and k >= 0.
%H A309896 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 A309896 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 A309896 Array starts:
%e A309896 [0] 1, 0, 0,  0,  0,  0,   0,   0,    0,    0,    0,    0, ...
%e A309896 [1] 1, 1, 1,  1,  1,  1,   1,   1,    1,    1,    1,    1, ...
%e A309896 [2] 1, 1, 2,  3,  5,  8,  13,  21,   34,   55,   89,  144, ...
%e A309896 [3] 1, 1, 3,  4,  9, 14,  28,  47,   89,  155,  286,  507, ...
%e A309896 [4] 1, 1, 4,  5, 14, 20,  48,  75,  165,  274,  571,  988, ...
%e A309896 [5] 1, 1, 5,  6, 20, 27,  75, 110,  275,  429, 1001, 1637, ...
%e A309896 [6] 1, 1, 6,  7, 27, 35, 110, 154,  429,  637, 1638, 2548, ...
%e A309896 [7] 1, 1, 7,  8, 35, 44, 154, 208,  637,  910, 2548, 3808, ...
%e A309896 [8] 1, 1, 8,  9, 44, 54, 208, 273,  910, 1260, 3808, 5508, ...
%e A309896 [9] 1, 1, 9, 10, 54, 65, 273, 350, 1260, 1700, 5508, 7752, ...
%o A309896 (SageMath)
%o A309896 @cached_function
%o A309896 def F(n, k):
%o A309896     if k <  0: return 0
%o A309896     if k == 0: return 1
%o A309896     a = sum((-1)^j*binomial(n-1-j,j  )*F(n,k-1-2*j) for j in (0..(n-1)/2))
%o A309896     b = sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
%o A309896     return a + b
%o A309896 print([F(n-k, k) for n in (0..11) for k in (0..n)])
%Y A309896 Cf. A000007 (n=0), A000012 (n=1), A000045 (n=2), A006053 (n=3), A188021 (n=4), A231181 (n=5).
%K A309896 nonn,tabl
%O A309896 0,13
%A A309896 _Peter Luschny_, Aug 21 2019