A374441 Triangle read by rows: T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling(k/2), floor(k/2)).

0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4, 0, 3, 0, 0, 0, 5, 0, 6, 0, 1, 0, 0, 6, 0, 10, 0, 4, 0, 0, 0, 7, 0, 15, 0, 10, 0, 1, 0, 0, 8, 0, 21, 0, 20, 0, 5, 0, 0, 0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 0, 0, 10, 0, 36, 0, 56, 0, 35, 0, 6, 0, 0, 0, 11, 0, 45, 0, 84, 0, 70, 0, 21, 0, 1, 0

Offset

0,8

Comments

Member of the family of Fibonacci polynomials (A011973, A162515, ...) and Chebyshev polynomials (A053119).

Links

Table of n, a(n) for n=0..90.

Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 257-271.

Formula

T(n, k) = [x^(n-k)][z^n] (x / (1 - x*z - z^2)).

T(n, k) = binomial(n - (k + 1)/2, (k + 1)/2) if k is odd, and otherwise 0.

Sum_{k=0..n} T(n, k) = Fibonacci(n + 1) - 1.

Columns with odd index agree with the odd indexed columns of A374440.

Example

Triangle starts:
  [ 0]  0;
  [ 1]  0, 0;
  [ 2]  0, 1, 0;
  [ 3]  0, 2, 0,  0;
  [ 4]  0, 3, 0,  1, 0;
  [ 5]  0, 4, 0,  3, 0,  0;
  [ 6]  0, 5, 0,  6, 0,  1, 0;
  [ 7]  0, 6, 0, 10, 0,  4, 0,  0;
  [ 8]  0, 7, 0, 15, 0, 10, 0,  1, 0;
  [ 9]  0, 8, 0, 21, 0, 20, 0,  5, 0, 0;
  [10]  0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 0;

Maple

T := (n, k) -> if k::even then 0 else binomial(n - (k + 1)/2, (k + 1)/2) fi:
# Or as a recurrence:
T := proc(n, k) option remember; if k::even or 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..12);

Crossrefs

Cf. A374440 (odd columns agree).

Cf. A000071 (row sums), A065941, A194005, A103631, A007318.

Cf. A011973, A162515, A053119.

Sequence in context: A173541 A333767 A325165 * A076849 A322977 A321374

Adjacent sequences: A374438 A374439 A374440 * A374442 A374443 A374444

Keyword

nonn,tabl,new

Author

Peter Luschny, Jul 19 2024

Status

approved