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, 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.

Links

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

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

Adjacent sequences: A374437 A374438 A374439 * A374441 A374442 A374443

Keyword

nonn,tabl

Author

Peter Luschny, Jul 21 2024

Status

approved