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

0,2

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A368892 A360727 A353013 * A316144 A138272 A367470

Adjacent sequences: A284753 A284754 A284755 * A284757 A284758 A284759

Keyword

nonn

Author

Seiichi Manyama, Apr 02 2017

Status

approved