A090598 a(n) = 5^(n-1/2) * (2*n-1)!! * Integral_{x = 0..1/2} 1/(1+x^2)^(n + 1/2) dx.

1, 14, 328, 10800, 458880, 23911680, 1477278720, 105623562240, 8582728089600, 781478859571200, 78834419151667200, 8729454895025356800, 1052840115930503577600, 137399767923711541248000, 19293267217416192000000000, 2900636848751631642132480000

Offset

1,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..324

Formula

From Robert Israel, Jun 27 2025: (Start)

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

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

(End)

a(n) ~ sqrt(Pi) * 10^(n - 1/2) * n^(n - 1/2) / exp(n). - Vaclav Kotesovec, Jun 28 2025

Maple

a:= n -> 5^(n - 1/2)*doublefactorial(2*n - 1)*int(1/(x^2 + 1)^(n + 1/2), x = 0 .. 1/2):
map(a, [$1..15]); # Robert Israel, Jun 27 2025

Mathematica

a[n_] := (5^(n - 1/2)(2n - 1)!!Integrate[1/(1 + x^2)^(n + 1/2), {x, 0, 1/2}]);
Table[a[n], {n, 1, 15}] (* Robert G. Wilson v, Feb 27 2004 *)

Crossrefs

Cf. A006882.

Sequence in context: A275397 A275262 A241373 * A277769 A060075 A222984

Adjacent sequences: A090595 A090596 A090597 * A090599 A090600 A090601

Keyword

nonn,frac

Author

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

Extensions

Edited and extended by Robert G. Wilson v, Feb 27 2004

Definition corrected by Robert Israel, Jun 27 2025

Status

approved