A190901 a(n) = Product_{k in M_n} k; M_n = {k | 1 <= k <= 2n and k mod 2 = n mod 2}.

1, 1, 8, 15, 384, 945, 46080, 135135, 10321920, 34459425, 3715891200, 13749310575, 1961990553600, 7905853580625, 1428329123020800, 6190283353629375, 1371195958099968000, 6332659870762850625, 1678343852714360832000, 8200794532637891559375, 2551082656125828464640000

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

Peter Luschny, Multifactorials.

Formula

a(2*k) = A006882(4*k) = 4^k * Gamma(2*k+1).

a(2*k+1) = A001147(2*k+1) = 4^k * Gamma(2*k+3/2) / sqrt(Pi/4).

From Amiram Eldar, Jul 21 2026: (Start)

a(n) ~ sqrt(Pi) * (2*n)^(n+1/2) / exp(n) for n even, and 2^(n+1/2) * n^n / exp(n) for n odd.

Sum_{n>=0} 1/a(n) = cosh(1/2) + (1/2)*sqrt(e*Pi/2)*erf(1/sqrt(2)) + (1/2)*sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erf and erfi are the error function and imaginary error function, respectively.

Sum_{n>=0} (-1)^n/a(n) = cosh(1/2) - (1/2)*sqrt(e*Pi/2)*erf(1/sqrt(2)) - (1/2)*sqrt(Pi/(2*e))*erfi(1/sqrt(2)). (End)

Maple

A190901 := proc(n) local k; mul(k, k = select(k-> k mod 2 = n mod 2, [$1 .. 2*n])) end: seq(A190901(n), n=0..18);

Mathematica

a[n_] := With[{m = Mod[n, 2]}, Product[If[Mod[k, 2] == m, k, 1], {k, 1, 2*n}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 27 2014 *)

Crossrefs

Cf. A001147, A006882.

Sequence in context: A212593 A153700 A343210 * A066916 A131446 A061746

Adjacent sequences: A190898 A190899 A190900 * A190902 A190903 A190904

Keyword

nonn,changed

Author

Peter Luschny, Jun 23 2011

Status

approved