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A374440
Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.
1
1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 1, 1, 1, 1, 4, 1, 3, 2, 0, 1, 5, 1, 6, 3, 1, 1, 1, 6, 1, 10, 4, 4, 3, 0, 1, 7, 1, 15, 5, 10, 6, 1, 1, 1, 8, 1, 21, 6, 20, 10, 5, 4, 0, 1, 9, 1, 28, 7, 35, 15, 15, 10, 1, 1, 1, 10, 1, 36, 8, 56, 21, 35, 20, 6, 5, 0
OFFSET
0,8
COMMENTS
Member of the family of Lucas-Fibonacci polynomials.
FORMULA
T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling((k + even(k) )/2), floor(k/2))) if k > 0, T(n, 0) = 1, where even(k) = 1 if k is even, otherwise 0.
Columns with odd index agree with the odd indexed columns of A374441.
EXAMPLE
Triangle starts:
[ 0] 1;
[ 1] 1, 0;
[ 2] 1, 1, 1;
[ 3] 1, 2, 1, 0;
[ 4] 1, 3, 1, 1, 1;
[ 5] 1, 4, 1, 3, 2, 0;
[ 6] 1, 5, 1, 6, 3, 1, 1;
[ 7] 1, 6, 1, 10, 4, 4, 3, 0;
[ 8] 1, 7, 1, 15, 5, 10, 6, 1, 1;
[ 9] 1, 8, 1, 21, 6, 20, 10, 5, 4, 0;
[10] 1, 9, 1, 28, 7, 35, 15, 15, 10, 1, 1;
MAPLE
T := proc(n, k) option remember; if k = 0 or k = 2 then 1 elif k > n then 0
elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9);
T := (n, k) -> ifelse(k = 0, 1, binomial(n - floor(k/2), ceil(k/2)) -
binomial(n - ceil((k + irem(k + 1, 2))/2), floor(k/2))):
CROSSREFS
Cf. A374441.
Cf. A000032 (Lucas), A001611 (even sums, Fibonacci + 1), A000071 (odd sums, Fibonacci - 1), A001911 (alternating sums, Fibonacci(n+3) - 2), A025560 (row lcm), A073028 (row max), A117671 & A025174 (central terms), A057979 (subdiagonal), A000217 (column 3).
Sequence in context: A156135 A047265 A341418 * A185962 A279928 A297325
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jul 21 2024
STATUS
approved