OFFSET
0,8
COMMENTS
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.
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);
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Jul 19 2024
STATUS
approved