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

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

Links

Table of n, a(n) for n=0..77.

Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

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: A103631 A263191 A192517 * A083856 A081718 A290353

Adjacent sequences: A309893 A309894 A309895 * A309897 A309898 A309899

Keyword

nonn,tabl

Author

Peter Luschny, Aug 21 2019

Status

approved