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A326303 Triangular array, read by rows: T(n,k) = numerator of Jtilde_k(n), 1 <= k <= 2*n+2.

%I A326303 #54 Oct 24 2019 11:34:45
%S A326303 1,1,2,3,1,1,8,41,65,11,1,1,16,147,13247,907,109,73,1,1,128,8649,
%T A326303 704707,1739,101717,3419,515,43,1,1,256,32307,660278641,6567221,
%U A326303 4557449,29273,76667,15389,251,67,1,1,1024,487889,357852111131,54281321,15689290781,151587391,115560397,1659311,254977,34061,1733,289,1,1
%N A326303 Triangular array, read by rows: T(n,k) = numerator of Jtilde_k(n), 1 <= k <= 2*n+2.
%C A326303 When a general definition was made in a recent paper, it was slightly different from the previous definition. Please check the annotation on page 15 of the paper in 2019.
%H A326303 Seiichi Manyama, <a href="/A326303/b326303.txt">Rows n = 0..15, flattened</a>
%H A326303 Kazufumi Kimoto, Masato Wakayama, <a href="http://doi.org/10.2206/kyushujm.60.383">Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators</a>, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).
%H A326303 Kazufumi Kimoto, Masato Wakayama, <a href="https://arxiv.org/abs/1905.01775">Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals</a>, arXiv:1905.01775 [math.PR], 2019. See p.22.
%F A326303 4*n^2 * Jtilde_k(n) = (8*n^2 - 8*n + 3) * Jtilde_k(n-1) - 4*(n - 1)^2 * Jtilde_k(n-2) + 4 * Jtilde_{k - 2}(n-1).
%F A326303 Jtilde_n(2*n+1) = Jtilde_n(2*n+2) = 1/A001044(n). So T(n,2*n+1) = T(n,2*n+2) = 1.
%e A326303 Triangle begins:
%e A326303       1,       1;
%e A326303     2/3,     3/4,          1,       1;
%e A326303    8/15,   41/64,      65/48,    11/8,     1/4,    1/4;
%e A326303   16/35, 147/256, 13247/8640, 907/576, 109/216, 73/144, 1/36, 1/36;
%o A326303 (Ruby)
%o A326303 def f(n)
%o A326303   return 1 if n < 2
%o A326303   (1..n).inject(:*)
%o A326303 end
%o A326303 def Jtilde(k, n)
%o A326303   return 0 if k == 0
%o A326303   return (2r ** n * f(n)) ** 2 / f(2 * n + 1) if k == 1
%o A326303   if n == 0
%o A326303     return 1 if k == 2
%o A326303     return 0
%o A326303   end
%o A326303   if n == 1
%o A326303     return 3r / 4 if k == 2
%o A326303     return 1      if k == 3 || k == 4
%o A326303     return 0
%o A326303   end
%o A326303   ((8r * n * n - 8 * n + 3) * Jtilde(k, n - 1) - 4 * (n - 1) ** 2 * Jtilde(k, n - 2) + 4 * Jtilde(k - 2, n - 1)) / (4 * n * n)
%o A326303 end
%o A326303 def A326303(n)
%o A326303   (0..n).map{|i| (1..2 * i + 2).map{|j| Jtilde(j, i).numerator}}.flatten
%o A326303 end
%o A326303 p A326303(10)
%Y A326303 Cf. A260832 (k=2), A264541 (k=3), A326748 (denominator).
%K A326303 nonn,frac,tabf
%O A326303 0,3
%A A326303 _Seiichi Manyama_, Oct 19 2019