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.
%I A373423 #19 Jun 12 2024 15:44:13 %S A373423 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, %T A373423 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, %U A373423 246,157,34,1,1,0,1,10,36,140,371,671,707,353,55,1,1,0 %N 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. %e A373423 Generating functions of row n: %e A373423 gf0 = 1; %e A373423 gf1 = - 1/( x-1); %e A373423 gf2 = x + 1/(-x+1); %e A373423 gf3 = x - 1/( x-1/( x+1)); %e A373423 gf4 = x + 1/(-x-1/(-x-1/(-x+1))); %e A373423 gf5 = x - 1/( x-1/( x-1/( x-1/( x+1)))); %e A373423 gf6 = x + 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x+1))))); %e A373423 . %e A373423 Array begins: %e A373423 [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, ... %e A373423 [1] 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A373423 [2] 1, 2, 1, 1, 1, 1, 1, 1, 1, ... A373565 %e A373423 [3] 1, 3, 3, 5, 8, 13, 21, 34, 55, ... A373566 %e A373423 [4] 1, 4, 6, 14, 31, 70, 157, 353, 793, ... A373567 %e A373423 [5] 1, 5, 10, 30, 85, 246, 707, 2037, 5864, ... A373568 %e A373423 [6] 1, 6, 15, 55, 190, 671, 2353, 8272, 29056, ... A373569 %e A373423 A000217, A006322, A108675, ... %e A373423 A000330, A085461, A244881, ... %e A373423 . %e A373423 Triangle starts: %e A373423 [0] 1; %e A373423 [1] 1, 0; %e A373423 [2] 1, 1, 0; %e A373423 [3] 1, 2, 1, 0; %e A373423 [4] 1, 3, 1, 1, 0; %e A373423 [5] 1, 4, 3, 1, 1, 0; %e A373423 [6] 1, 5, 6, 5, 1, 1, 0; %p A373423 row := proc(n, len) local x, a, j, ser; %p A373423 if n = 0 then a := -1 elif n = 1 then a := -1/(x - 1) elif irem(n, 2) = 1 then %p A373423 a := x + 1; for j from 1 to n-1 do a := x - 1 / a od: else %p A373423 a := -x + 1; for j from 1 to n-1 do a := -x - 1 / a od: fi; %p A373423 ser := series((-1)^(n-1)*a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end: %p A373423 A := (n, k) -> row(n, 12)[k+1]: # array form %p A373423 T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form %p A373423 seq(lprint([n], row(n, 9)), n = 0..9); %o A373423 (SageMath) %o A373423 def Arow(n, len): %o A373423 R.<x> = PowerSeriesRing(ZZ, len) %o A373423 if n == 0: return [1] + [0]*(len - 1) %o A373423 if n == 1: return [1]*(len - 1) %o A373423 x = x if n % 2 == 1 else -x %o A373423 a = x + 1 %o A373423 for _ in range(n - 1): %o A373423 a = x - 1 / a %o A373423 if n % 2 == 0: a = -a %o A373423 return a.list() %o A373423 for n in range(8): print(Arow(n, 9)) %Y A373423 Cf. A373424, A276312 (main diagonal). %Y A373423 Rows include: A373565, A373566, A373567, A373568, A373569. %Y A373423 Columns include: A000217, A000330, A006322, A085461, A108675, A244881. %K A373423 nonn,tabl %O A373423 0,8 %A A373423 _Peter Luschny_, Jun 09 2024