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A309896
Generalized Fibonacci numbers. Square array read by ascending antidiagonals. F(n,k) for n >= 0 and k >= 0.
3
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, 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, 34, 1, 0, 1, 1, 9, 9, 35, 35, 75, 75, 89, 55, 1, 0
OFFSET
0,13
FORMULA
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.
EXAMPLE
Array starts:
[0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
[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, ...
[3] 1, 1, 3, 4, 9, 14, 28, 47, 89, 155, 286, 507, ...
[4] 1, 1, 4, 5, 14, 20, 48, 75, 165, 274, 571, 988, ...
[5] 1, 1, 5, 6, 20, 27, 75, 110, 275, 429, 1001, 1637, ...
[6] 1, 1, 6, 7, 27, 35, 110, 154, 429, 637, 1638, 2548, ...
[7] 1, 1, 7, 8, 35, 44, 154, 208, 637, 910, 2548, 3808, ...
[8] 1, 1, 8, 9, 44, 54, 208, 273, 910, 1260, 3808, 5508, ...
[9] 1, 1, 9, 10, 54, 65, 273, 350, 1260, 1700, 5508, 7752, ...
PROG
(SageMath)
@cached_function
def F(n, k):
if k < 0: return 0
if k == 0: return 1
a = sum((-1)^j*binomial(n-1-j, j )*F(n, k-1-2*j) for j in (0..(n-1)/2))
b = sum((-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for j in (0..(n-2)/2))
return a + b
print([F(n-k, k) for n in (0..11) for k in (0..n)])
CROSSREFS
Cf. A000007 (n=0), A000012 (n=1), A000045 (n=2), A006053 (n=3), A188021 (n=4), A231181 (n=5).
Sequence in context: A374176 A263191 A192517 * A083856 A081718 A290353
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Aug 21 2019
STATUS
approved