A373424 - OEIS

A373424

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

%I #17 Jun 13 2024 01:57:48

%S 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,5,1,0,1,5,10,14,8,1,0,1,6,15,30,

%T 31,13,1,0,1,7,21,55,85,70,21,1,0,1,8,28,91,190,246,157,34,1,0,1,9,36,

%U 140,371,671,707,353,55,1,0,1,10,45,204,658,1547,2353,2037,793,89,1,0

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

%C A variant of both A050446 and A050447 which are the main entries. Differs in indexing and adds a first row to the array resp. a diagonal to the triangle.

%H T. Kyle Petersen and Yan Zhuang, <a href="https://arxiv.org/abs/2403.07181">Zig-zag Eulerian polynomials</a>, arXiv:2403.07181 [math.CO], 2024. (Table 3)

%e Generating functions of the rows:

%e gf0 = 1;

%e gf1 = -1/( x-1);

%e gf2 = 1/(-x-1/(-x-1));

%e gf3 = -1/( x-1/( x-1/( x-1)));

%e gf4 = 1/(-x-1/(-x-1/(-x-1/(-x-1))));

%e gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));

%e gf6 = 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));

%e ...

%e Array A(n, k) starts:

%e [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007

%e [1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012

%e [2] 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... A000045

%e [3] 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, ... A006356

%e [4] 1, 4, 10, 30, 85, 246, 707, 2037, 5864, 16886, ... A006357

%e [5] 1, 5, 15, 55, 190, 671, 2353, 8272, 29056, 102091, ... A006358

%e [6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ... A006359

%e A000027,A000330, A085461, A244881, ...

%e A000217, A006322, A108675, ...

%e .

%e Triangle T(n, k) = A(n - k, k) starts:

%e [0] 1;

%e [1] 1, 0;

%e [2] 1, 1, 0;

%e [3] 1, 2, 1, 0;

%e [4] 1, 3, 3, 1, 0;

%e [5] 1, 4, 6, 5, 1, 0;

%e [6] 1, 5, 10, 14, 8, 1, 0;

%p row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then

%p a := x - 1; for j from 1 to n do a := x - 1 / a od: a := a - x; else

%p a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;

%p fi; ser := series(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

%o (SageMath)

%o def Arow(n, len):

%o R.<x> = PowerSeriesRing(ZZ, len)

%o if n == 0: return [1] + [0]*(len - 1)

%o x = -x if n % 2 else x

%o a = x + 1

%o for _ in range(n):

%o a = x - 1 / a

%o a = x - a if n % 2 else a - x

%o return a.list()

%o for n in range(7): print(Arow(n, 10))

%Y Cf. A050446, A050447, A276313 (main diagonal), A373353 (row sums of triangle).

%Y Cf. A373423.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Jun 09 2024