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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.

%I A373424 #17 Jun 13 2024 01:57:48
%S A373424 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 A373424 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 A373424 140,371,671,707,353,55,1,0,1,10,45,204,658,1547,2353,2037,793,89,1,0
%N 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.
%C A373424 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 A373424 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 A373424 Generating functions of the rows:
%e A373424    gf0 =  1;
%e A373424    gf1 = -1/( x-1);
%e A373424    gf2 =  1/(-x-1/(-x-1));
%e A373424    gf3 = -1/( x-1/( x-1/( x-1)));
%e A373424    gf4 =  1/(-x-1/(-x-1/(-x-1/(-x-1))));
%e A373424    gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));
%e A373424    gf6 =  1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));
%e A373424    ...
%e A373424 Array A(n, k) starts:
%e A373424   [0] 1, 0,  0,  0,   0,    0,    0,     0,      0,      0, ...  A000007
%e A373424   [1] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ...  A000012
%e A373424   [2] 1, 2,  3,  5,   8,   13,   21,    34,     55,     89, ...  A000045
%e A373424   [3] 1, 3,  6, 14,  31,   70,  157,   353,    793,   1782, ...  A006356
%e A373424   [4] 1, 4, 10, 30,  85,  246,  707,  2037,   5864,  16886, ...  A006357
%e A373424   [5] 1, 5, 15, 55, 190,  671, 2353,  8272,  29056, 102091, ...  A006358
%e A373424   [6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ...  A006359
%e A373424    A000027,A000330,   A085461,     A244881, ...
%e A373424        A000217, A006322,    A108675, ...
%e A373424 .
%e A373424 Triangle T(n, k) = A(n - k, k) starts:
%e A373424   [0] 1;
%e A373424   [1] 1,  0;
%e A373424   [2] 1,  1,  0;
%e A373424   [3] 1,  2,  1,  0;
%e A373424   [4] 1,  3,  3,  1,  0;
%e A373424   [5] 1,  4,  6,  5,  1,  0;
%e A373424   [6] 1,  5, 10, 14,  8,  1, 0;
%p A373424 row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then
%p A373424 a :=  x - 1; for j from 1 to n do a :=  x - 1 / a od: a :=  a - x; else
%p A373424 a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;
%p A373424 fi; ser := series(a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
%p A373424 A := (n, k) -> row(n, 12)[k+1]:      # array form
%p A373424 T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form
%o A373424 (SageMath)
%o A373424 def Arow(n, len):
%o A373424     R.<x> = PowerSeriesRing(ZZ, len)
%o A373424     if n == 0: return [1] + [0]*(len - 1)
%o A373424     x = -x if n % 2 else x
%o A373424     a = x + 1
%o A373424     for _ in range(n):
%o A373424         a = x - 1 / a
%o A373424     a = x - a if n % 2 else a - x
%o A373424     return a.list()
%o A373424 for n in range(7): print(Arow(n, 10))
%Y A373424 Cf. A050446, A050447, A276313 (main diagonal), A373353 (row sums of triangle).
%Y A373424 Cf. A373423.
%K A373424 nonn,tabl
%O A373424 0,8
%A A373424 _Peter Luschny_, Jun 09 2024