A284756 (n + 1)^3*a(n + 1) = 2*(2*n + 1)*(5*n^2 + 5*n + 2)*a(n) - 8*n*(7*n^2 + 1)*a(n - 1) + 22*n*(n - 1)*(2*n - 1)*a(n - 2), with a(0) = 1, a(1) = 4 and a(2) = 28.

1, 4, 28, 268, 3004, 36784, 476476, 6418192, 88986172, 1261473136, 18200713168, 266393373712, 3945664966204, 59029237351504, 890697897694192, 13539585443232688, 207149418061499452, 3187355160332835184, 49290960047575223824, 765703166164798253392

Offset: 0

Author

Seiichi Manyama, Apr 02 2017

Keywords

• nonn

Comments

This sequence is c_11 in the 2015 paper of Cooper et al.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..819
• S. Cooper, Ramanujan's theories of elliptic functions to alternative bases, and beyond, talk slides, Askey 80 Conference, 2013.
• S. Cooper, J. Ge and D. Ye, Hypergeometric transformation formulas of degrees 3, 7, 11 and 23, Journal of Mathematical Analysis and Applications, Volume 421, Issue 2, 15 January 2015, Pages 1358-1376.

Crossrefs

Cf. A184423 (c_3), A183204 (c_7), this sequence (c_11).

Programs

Mathematica

RecurrenceTable[{(n+1)^3*a[n+1] == 2*(2*n+1)*(5*n^2+5*n+2)*a[n] - 8*n*(7*n^2+1)*a[n-1] + 22*n*(n-1)*(2*n-1)*a[n-2], a[0]==1, a[1]==4, a[2]==28}, a, {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2017 *)

Ruby

def A284756(n)
  a, b, c, i = 0, 0, 1, -1
  ary = [0, 0]
  while i < n
    i += 1
    j = 2 * (2 * i + 1) * (5 * i * i + 5 * i + 2) * c - 8 * i * (7 * i * i + 1) * b + 22 * i * (i - 1) * (2 * i - 1) * a
    break if j % ((i + 1) ** 3) > 0
    a, b, c = b, c, j / ((i + 1) ** 3)
    ary << b
  end
  ary[2..-1]
end

Formula

a(n) ~ c * d^n / (Pi*n)^(3/2), where d = 16.8275008141470347474718307386716769... is the real root of the equation -44 + 56*d - 20*d^2 + d^3 = 0 and c = 1.83051467150137478416073409831908489312609... is the positive real root of the equation -1331 - 1020*c^2 - 1936*c^4 + 704*c^6 = 0. - Vaclav Kotesovec, Apr 02 2017