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%I #38 Apr 02 2017 10:02:22
%S 1,4,28,268,3004,36784,476476,6418192,88986172,1261473136,18200713168,
%T 266393373712,3945664966204,59029237351504,890697897694192,
%U 13539585443232688,207149418061499452,3187355160332835184,49290960047575223824,765703166164798253392
%N (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.
%C This sequence is c_11 in the 2015 paper of Cooper et al.
%H Seiichi Manyama, <a href="/A284756/b284756.txt">Table of n, a(n) for n = 0..819</a>
%H S. Cooper, <a href="http://www-users.math.umn.edu/~stant001/ASKEYABS/Shaun_Cooper">Ramanujan's theories of elliptic functions to alternative bases, and beyond</a>, talk slides, Askey 80 Conference, 2013.
%H S. Cooper, J. Ge and D. Ye, <a href="http://dx.doi.org/10.1016/j.jmaa.2014.07.061">Hypergeometric transformation formulas of degrees 3, 7, 11 and 23</a>, Journal of Mathematical Analysis and Applications, Volume 421, Issue 2, 15 January 2015, Pages 1358-1376.
%F 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
%t 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 *)
%o (Ruby)
%o def A284756(n)
%o a, b, c, i = 0, 0, 1, -1
%o ary = [0, 0]
%o while i < n
%o i += 1
%o 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
%o break if j % ((i + 1) ** 3) > 0
%o a, b, c = b, c, j / ((i + 1) ** 3)
%o ary << b
%o end
%o ary[2..-1]
%o end
%Y Cf. A184423 (c_3), A183204 (c_7), this sequence (c_11).
%K nonn
%O 0,2
%A _Seiichi Manyama_, Apr 02 2017
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