A373423 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(0) = 1, cf(1) = -1/(x - 1), and for n > 1 is cf(n) = ~( ~x - 1/(~x - 1/(~x - 1/(~x - 1/(~x - ... 1/(~x + 1))))...) ) where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 5, 6, 5, 1, 1, 0, 1, 6, 10, 14, 8, 1, 1, 0, 1, 7, 15, 30, 31, 13, 1, 1, 0, 1, 8, 21, 55, 85, 70, 21, 1, 1, 0, 1, 9, 28, 91, 190, 246, 157, 34, 1, 1, 0, 1, 10, 36, 140, 371, 671, 707, 353, 55, 1, 1, 0

Offset

0,8

Links

Table of n, a(n) for n=0..77.

Example

Generating functions of row n:
   gf0 = 1;
   gf1 =   - 1/( x-1);
   gf2 = x + 1/(-x+1);
   gf3 = x - 1/( x-1/( x+1));
   gf4 = x + 1/(-x-1/(-x-1/(-x+1)));
   gf5 = x - 1/( x-1/( x-1/( x-1/( x+1))));
   gf6 = x + 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x+1)))));
.
Array begins:
  [0] 1, 0,  0,   0,   0,    0,     0,     0,      0, ...
  [1] 1, 1,  1,   1,   1,    1,     1,     1,      1, ...
  [2] 1, 2,  1,   1,   1,    1,     1,     1,      1, ...  A373565
  [3] 1, 3,  3,   5,   8,   13,    21,    34,     55, ...  A373566
  [4] 1, 4,  6,  14,  31,   70,   157,   353,    793, ...  A373567
  [5] 1, 5, 10,  30,  85,  246,   707,  2037,   5864, ...  A373568
  [6] 1, 6, 15,  55, 190,  671,  2353,  8272,  29056, ...  A373569
       A000217,  A006322,     A108675, ...
            A000330,   A085461,      A244881, ...
.
Triangle starts:
  [0] 1;
  [1] 1, 0;
  [2] 1, 1,  0;
  [3] 1, 2,  1,  0;
  [4] 1, 3,  1,  1,  0;
  [5] 1, 4,  3,  1,  1,  0;
  [6] 1, 5,  6,  5,  1,  1, 0;

Maple

row := proc(n, len) local x, a, j, ser;
if n = 0 then a := -1 elif n = 1 then a := -1/(x - 1) elif irem(n, 2) = 1 then
  a :=  x + 1; for j from 1 to n-1 do a :=  x - 1 / a od: else
  a := -x + 1; for j from 1 to n-1 do a := -x - 1 / a od: fi;
ser := series((-1)^(n-1)*a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]:      # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form
seq(lprint([n], row(n, 9)), n = 0..9);

Prog

(SageMath)
def Arow(n, len):
    R.<x> = PowerSeriesRing(ZZ, len)
    if n == 0: return [1] + [0]*(len - 1)
    if n == 1: return [1]*(len - 1)
    x = x if n % 2 == 1 else -x
    a = x + 1
    for _ in range(n - 1):
        a = x - 1 / a
    if n % 2 == 0: a = -a
    return a.list()
for n in range(8): print(Arow(n, 9))

Crossrefs

Cf. A373424, A276312 (main diagonal).

Rows include: A373565, A373566, A373567, A373568, A373569.

Columns include: A000217, A000330, A006322, A085461, A108675, A244881.

Sequence in context: A089263 A348951 A369815 * A156135 A047265 A341418

Adjacent sequences: A373420 A373421 A373422 * A373424 A373425 A373426

Keyword

nonn,tabl

Author

Peter Luschny, Jun 09 2024

Status

approved