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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.
2
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, 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, 140, 371, 671, 707, 353, 55, 1, 0, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 0
OFFSET
0,8
COMMENTS
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.
LINKS
T. Kyle Petersen and Yan Zhuang, Zig-zag Eulerian polynomials, arXiv:2403.07181 [math.CO], 2024. (Table 3)
EXAMPLE
Generating functions of the rows:
gf0 = 1;
gf1 = -1/( x-1);
gf2 = 1/(-x-1/(-x-1));
gf3 = -1/( x-1/( x-1/( x-1)));
gf4 = 1/(-x-1/(-x-1/(-x-1/(-x-1))));
gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));
gf6 = 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));
...
Array A(n, k) starts:
[0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007
[1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
[2] 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... A000045
[3] 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, ... A006356
[4] 1, 4, 10, 30, 85, 246, 707, 2037, 5864, 16886, ... A006357
[5] 1, 5, 15, 55, 190, 671, 2353, 8272, 29056, 102091, ... A006358
[6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ... A006359
.
Triangle T(n, k) = A(n - k, k) starts:
[0] 1;
[1] 1, 0;
[2] 1, 1, 0;
[3] 1, 2, 1, 0;
[4] 1, 3, 3, 1, 0;
[5] 1, 4, 6, 5, 1, 0;
[6] 1, 5, 10, 14, 8, 1, 0;
MAPLE
row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then
a := x - 1; for j from 1 to n do a := x - 1 / a od: a := a - x; else
a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;
fi; ser := series(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
PROG
(SageMath)
def Arow(n, len):
R.<x> = PowerSeriesRing(ZZ, len)
if n == 0: return [1] + [0]*(len - 1)
x = -x if n % 2 else x
a = x + 1
for _ in range(n):
a = x - 1 / a
a = x - a if n % 2 else a - x
return a.list()
for n in range(7): print(Arow(n, 10))
CROSSREFS
Cf. A050446, A050447, A276313 (main diagonal), A373353 (row sums of triangle).
Cf. A373423.
Sequence in context: A349841 A339779 A373449 * A277504 A167763 A277666
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jun 09 2024
STATUS
approved