A090598 - OEIS

%I #24 Jun 29 2025 09:02:44

%S 1,14,328,10800,458880,23911680,1477278720,105623562240,8582728089600,

%T 781478859571200,78834419151667200,8729454895025356800,

%U 1052840115930503577600,137399767923711541248000,19293267217416192000000000,2900636848751631642132480000

%N a(n) = 5^(n-1/2) * (2*n-1)!! * Integral_{x = 0..1/2} 1/(1+x^2)^(n + 1/2) dx.

%C Former name was "Numerator of ((integral_{x = 0..1/2} 1/(1+x^2)^(n + 1/2) dx) * sqrt(1/5))."

%H Robert Israel, <a href="/A090598/b090598.txt">Table of n, a(n) for n = 1..324</a>

%F From _Robert Israel_, Jun 27 2025: (Start)

%F a(n) = 5^(n - 1/2) * (2*n - 1)!! * hypergeom([1/2, n + 1/2], [3/2], -1/4)/2.

%F a(n + 2) = (-80*n^2 - 40*n)*a(n) + (18*n + 14)*a(n + 1).

%F (End)

%F a(n) ~ sqrt(Pi) * 10^(n - 1/2) * n^(n - 1/2) / exp(n). - _Vaclav Kotesovec_, Jun 28 2025

%p a:= n -> 5^(n - 1/2)*doublefactorial(2*n - 1)*int(1/(x^2 + 1)^(n + 1/2), x = 0 .. 1/2):

%p map(a, [$1..15]); # _Robert Israel_, Jun 27 2025

%t a[n_] := (5^(n - 1/2)(2n - 1)!!Integrate[1/(1 + x^2)^(n + 1/2), {x, 0, 1/2}]);

%t Table[a[n], {n, 1, 15}] (* _Robert G. Wilson v_, Feb 27 2004 *)

%Y Cf. A006882.

%K nonn,frac

%O 1,2

%A Al Hakanson (hawkuu(AT)excite.com), Feb 25 2004

%E Edited and extended by _Robert G. Wilson v_, Feb 27 2004

%E Definition corrected by _Robert Israel_, Jun 27 2025