A366987 - OEIS

%I #68 Nov 07 2023 08:23:10

%S -1,0,0,-2,-1,-2,2,1,-1,-2,-6,-3,-3,-3,-6,10,5,1,-1,-5,-10,-22,-11,-7,

%T -5,-7,-11,-22,42,21,9,3,-3,-9,-21,-42,-86,-43,-23,-13,-11,-13,-23,

%U -43,-86,170,85,41,19,5,-5,-19,-41,-85,-170,-342,-171,-87,-45,-27,-21,-27,-45,-87,-171,-342

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

%H Paolo Xausa, <a href="/A366987/b366987.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened)

%F T(n, 0) = -((-2)^n + 2)/3.

%F T(n, k+1) - T(n, k) = T(n-1, k) + (-1)^k.

%F T(2*n+1, n) = A001045(n).

%F T(2*n+1, n+1) = -A001045(n).

%F T(2*n, n+1) = -A048573(n-1), for n > 0.

%F Note that the definition of T extends to negative parameters:

%F T(2*n-2, n-1) = -A001045(n).

%F -2^n*Sum_{k=0..n} (-1)^k*T(-n, -k) = A059570(n+1).

%F -4^n*Sum_{k=0..2*n} T(-2*n, -k) = A020989(n).

%F -Sum_{k=0..n} (-1)^k*T(n, k) = A077898(n). See also A053088.

%F Sum_{k = 0..2*n} |T(2*n, k)| = (4^(n+1) - 1)/3.

%F Sum_{k = 0..2*n+1} |T(2*n+1, k)| = (1 + (-1)^n - 2^(2 + n) + 2^(1 + 2*n))/3.

%F G.f.: (-1 - x + x*y)/((1 - x)*(1 + 2*x)*(1 + x*y)*(1 - 2*x*y)). - _Stefano Spezia_, Nov 03 2023

%e Triangle T(n, k) starts:

%e    -1

%e     0   0

%e    -2  -1  -2

%e     2   1  -1  -2

%e    -6  -3  -3  -3  -6

%e    10   5   1  -1  -5 -10

%e   -22 -11  -7  -5  -7 -11 -22

%e    42  21   9   3  -3  -9 -21 -42

%e    ...

%e Note the symmetrical distribution of the absolute values of the terms in each row.

%p T := (n, k) -> -(2^(n-k)*(-1)^n + 2^k + (-1)^k)/3:

%p seq(seq(T(n, k), k = 0..n), n = 0..10);  # _Peter Luschny_, Nov 02 2023

%t 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 *)

%o (PARI) T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ _Thomas Scheuerle_, Nov 01 2023

%Y Rows sums: -A282577(n+2), if the conjectures from _Colin Barker_ in A282577 are true.

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

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

%Y Third column: -A140966(n).

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

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

%K sign,tabl

%O 0,4

%A _Paul Curtz_ and _Thomas Scheuerle_, Oct 31 2023

%E a(42) corrected by _Paolo Xausa_, Nov 07 2023