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