A374440 - OEIS

%I #12 Jul 22 2024 08:19:03

%S 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,

%T 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,

%U 10,1,1,1,10,1,36,8,56,21,35,20,6,5,0

%N 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.

%C Member of the family of Lucas-Fibonacci polynomials.

%F 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.

%F Columns with odd index agree with the odd indexed columns of A374441.

%e Triangle starts:

%e   [ 0]  1;

%e   [ 1]  1,  0;

%e   [ 2]  1,  1,  1;

%e   [ 3]  1,  2,  1,  0;

%e   [ 4]  1,  3,  1,  1,  1;

%e   [ 5]  1,  4,  1,  3,  2,  0;

%e   [ 6]  1,  5,  1,  6,  3,  1,  1;

%e   [ 7]  1,  6,  1, 10,  4,  4,  3,  0;

%e   [ 8]  1,  7,  1, 15,  5, 10,  6,  1,  1;

%e   [ 9]  1,  8,  1, 21,  6, 20, 10,  5,  4,  0;

%e   [10]  1,  9,  1, 28,  7, 35, 15, 15, 10,  1, 1;

%p T := proc(n, k) option remember; if k = 0 or k = 2 then 1 elif k > n then 0

%p elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

%p T := (n, k) -> ifelse(k = 0, 1, binomial(n - floor(k/2), ceil(k/2)) -

%p binomial(n - ceil((k + irem(k + 1, 2))/2), floor(k/2))):

%Y Cf. A374441.

%Y 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).

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Jul 21 2024