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

Author

Peter Luschny, Jul 19 2024

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
• Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.

Crossrefs

Cf. A374440 (odd columns agree).

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

Cf. A011973, A162515, A053119.

Programs

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);

Mathematica

A374441[n_, k_] := If[OddQ[k], Binomial[n - (k + 1)/2, (k + 1)/2], 0];
Table[A374441[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Nov 16 2024 *)

Python

from math import isqrt, comb
def A374441(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-comb(a+1,2)
    return comb(a-(b+1>>1),b+1>>1) if b&1 else 0 # Chai Wah Wu, Nov 14 2024

Python

from math import comb as binomial
def row(n: int) -> list[int]:
    return [binomial(n - (k+1)//2, (k+1)//2) if k%2 else 0 for k in range(n+1)]
for n in range(11): print(row(n))  # Peter Luschny, Nov 21 2024

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.