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
Examples
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;
Crossrefs
Programs
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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);
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SageMath
def Arow(n, len): R.= 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))