A366987 Triangle read by rows: T(n, k) = -(2^(n - k)*(-1)^n + 2^k + (-1)^k)/3.

-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, -86, -43, -23, -13, -11, -13, -23, -43, -86, 170, 85, 41, 19, 5, -5, -19, -41, -85, -170, -342, -171, -87, -45, -27, -21, -27, -45, -87, -171, -342

Offset: 0

Author

Paul Curtz and Thomas Scheuerle, Oct 31 2023

Keywords

• sign
• tabl

Examples

			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.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened)

Crossrefs

Rows sums: -A282577(n+2), if the conjectures from Colin Barker in A282577 are true.

First column: -(-1)^n * A078008(n).

Second column: (-1)^n * A001045(n).

Third column: -A140966(n).

Fourth column: (-1)^n * A155980(n+2).

Cf. A000079, A020989, A048573, A053088, A059570, A077898, A140966.

Programs

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 *)

PARI

T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ Thomas Scheuerle, Nov 01 2023

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..n} (-1)^k*T(n, k) = A077898(n). See also A053088.

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

Extensions

a(42) corrected by Paolo Xausa, Nov 07 2023