This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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;
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);
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))