A380909 a(n) = numerator(n!! / (n - 1)!!).

1, 1, 2, 3, 8, 15, 16, 35, 128, 315, 256, 693, 1024, 3003, 2048, 6435, 32768, 109395, 65536, 230945, 262144, 969969, 524288, 2028117, 4194304, 16900975, 8388608, 35102025, 33554432, 145422675, 67108864, 300540195, 2147483648, 9917826435, 4294967296, 20419054425

Offset: 0

Author

Peter Luschny, Feb 09 2025

Keywords

• nonn
• frac

Comments

Perhaps surprisingly, the double factorial is also defined for n = -1. The reason for this is that (-1/2)! is well defined. See the first formula.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Gamma Function, Digital library of mathematical functions, Feb. 2024.

Crossrefs

Cf. A380910 (denominator), A006882, A095987, A004730, A004731.

Programs

Maple

seq(numer(doublefactorial(n) / doublefactorial(n - 1)), n = 0..20);
# Alternative:
a := n -> numer((GAMMA(n/2 + 1) / GAMMA(n/2 + 1/2)) * ifelse(n::even, sqrt(Pi), 2/sqrt(Pi))):
seq(a(n), n = 0..35);

Mathematica

A380909[n_] := Numerator[n!!/(n - 1)!!]; Array[A380909, 50, 0] (* or *)
Numerator[FoldList[#2/# &, 1, Range[49]]] (* Paolo Xausa, Feb 11 2025 *)

Python

from fractions import Fraction
from functools import cache
@cache
def R(n: int) -> Fraction:
    if n == 0: return Fraction(1, 1)
    return Fraction(n, 1) / R(n - 1)
def aList(upto:int) -> list[int]:
    return [R(n).numerator for n in range(upto + 1)]
print(aList(35))

Formula

r(n) = ((n/2)! / ((n - 1) / 2)!) * [sqrt(Pi) if n is even otherwise 2/sqrt(Pi)].

r(n) = n / r(n - 1) for n >= 1.

r(n) ~ sqrt(n)*exp(1/(4*n))*(Pi/2)^(cos(Pi*n)/2).

Product_{k=0..n} r(k) = A006882(n).

a(n) = numerator(r(n)).

a(n) = A006882(n)/A095987(n). - R. J. Mathar, Feb 10 2025

a(n) = A004731(n+1). - R. J. Mathar, Feb 10 2025