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

1, 1, 2, 3, 1, 1, 8, 41, 65, 11, 1, 1, 16, 147, 13247, 907, 109, 73, 1, 1, 128, 8649, 704707, 1739, 101717, 3419, 515, 43, 1, 1, 256, 32307, 660278641, 6567221, 4557449, 29273, 76667, 15389, 251, 67, 1, 1, 1024, 487889, 357852111131, 54281321, 15689290781, 151587391, 115560397, 1659311, 254977, 34061, 1733, 289, 1, 1

Offset: 0

Author

Seiichi Manyama, Oct 19 2019

Keywords

• nonn
• frac
• tabf

Comments

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.

Examples

			Triangle begins:
      1,       1;
    2/3,     3/4,          1,       1;
   8/15,   41/64,      65/48,    11/8,     1/4,    1/4;
  16/35, 147/256, 13247/8640, 907/576, 109/216, 73/144, 1/36, 1/36;
		

Links

• Seiichi Manyama, Rows n = 0..15, flattened
• Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).
• Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals, arXiv:1905.01775 [math.PR], 2019. See p.22.

Crossrefs

Cf. A260832 (k=2), A264541 (k=3), A326748 (denominator).

Programs

Ruby

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def Jtilde(k, n)
  return 0 if k == 0
  return (2r ** n * f(n)) ** 2 / f(2 * n + 1) if k == 1
  if n == 0
    return 1 if k == 2
    return 0
  end
  if n == 1
    return 3r / 4 if k == 2
    return 1      if k == 3 || k == 4
    return 0
  end
  ((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)
end
def A326303(n)
  (0..n).map{|i| (1..2 * i + 2).map{|j| Jtilde(j, i).numerator}}.flatten
end
p A326303(10)

Formula

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).

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.