OFFSET
0,4
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened)
FORMULA
T(n, 0) = -((-2)^n + 2)/3.
T(n, k+1) - T(n, k) = T(n-1, k) + (-1)^k.
T(2*n+1, n) = A001045(n).
T(2*n+1, n+1) = -A001045(n).
T(2*n, n+1) = -A048573(n-1), for n > 0.
Note that the definition of T extends to negative parameters:
T(2*n-2, n-1) = -A001045(n).
-2^n*Sum_{k=0..n} (-1)^k*T(-n, -k) = A059570(n+1).
-4^n*Sum_{k=0..2*n} T(-2*n, -k) = A020989(n).
Sum_{k = 0..2*n} |T(2*n, k)| = (4^(n+1) - 1)/3.
Sum_{k = 0..2*n+1} |T(2*n+1, k)| = (1 + (-1)^n - 2^(2 + n) + 2^(1 + 2*n))/3.
G.f.: (-1 - x + x*y)/((1 - x)*(1 + 2*x)*(1 + x*y)*(1 - 2*x*y)). - Stefano Spezia, Nov 03 2023
EXAMPLE
Triangle T(n, k) starts:
-1
0 0
-2 -1 -2
2 1 -1 -2
-6 -3 -3 -3 -6
10 5 1 -1 -5 -10
-22 -11 -7 -5 -7 -11 -22
42 21 9 3 -3 -9 -21 -42
...
Note the symmetrical distribution of the absolute values of the terms in each row.
MAPLE
T := (n, k) -> -(2^(n-k)*(-1)^n + 2^k + (-1)^k)/3:
seq(seq(T(n, k), k = 0..n), n = 0..10); # Peter Luschny, Nov 02 2023
MATHEMATICA
A366987row[n_]:=Table[-(2^(n-k)(-1)^n+2^k+(-1)^k)/3, {k, 0, n}]; Array[A366987row, 15, 0] (* Paolo Xausa, Nov 07 2023 *)
PROG
(PARI) T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ Thomas Scheuerle, Nov 01 2023
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul Curtz and Thomas Scheuerle, Oct 31 2023
EXTENSIONS
a(42) corrected by Paolo Xausa, Nov 07 2023
STATUS
approved