A326748 Triangular array, read by rows: T(n,k) = denominator of Jtilde_k(n), 1 <= k <= 2*n+2.

1, 1, 3, 4, 1, 1, 15, 64, 48, 8, 4, 4, 35, 256, 8640, 576, 216, 144, 36, 36, 315, 16384, 430080, 1024, 138240, 4608, 6912, 576, 576, 576, 693, 65536, 387072000, 3686400, 4838400, 30720, 576000, 115200, 43200, 11520, 14400, 14400

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. A056982 (k=2), A264542(n)/2 (k=3) (By the definition of A264542, Jtilde3(1)(1) = 1/2).

Cf. A001044, A326303 (numerator).

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 A326748(n)
  (0..n).map{|i| (1..2 * i + 2).map{|j| Jtilde(j, i).denominator}}.flatten
end
p A326748(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) = A001044(n).