A380909 -id:A380909 - cp's OEIS frontend

A380910 a(n) = denominator(n!! / (n - 1)!!).

1, 1, 1, 2, 3, 8, 5, 16, 35, 128, 63, 256, 231, 1024, 429, 2048, 6435, 32768, 12155, 65536, 46189, 262144, 88179, 524288, 676039, 4194304, 1300075, 8388608, 5014575, 33554432, 9694845, 67108864, 300540195, 2147483648, 583401555, 4294967296, 2268783825, 17179869184

Offset: 0

Peter Luschny, Feb 09 2025

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

Cf. A380909 (numerator).

Cf. A004730, A004731.

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

A380910[n_] := Denominator[n!!/(n - 1)!!]; Array[A380910, 50, 0] (* or *)
Denominator[FoldList[#2/# &, 1, Range[49]]] (* Paolo Xausa, Feb 11 2025 *)

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).denominator for n in range(upto + 1)]
print(aList(37))

r(n) = ((n/2)! / ((n - 1) / 2)!) * [sqrt(Pi) if n is even otherwise 2/sqrt(Pi)].
a(n) = denominator(r(n)).
a(n) = A004730(n-1). - R. J. Mathar, Feb 10 2025
a(n) = A006882(n-1)/A095987(n). - R. J. Mathar, Feb 10 2025

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

Original entry on oeis.org

0, 1, 1, 4, 9, 64, 75, 256, 1225, 16384, 19845, 65536, 160083, 1048576, 1288287, 4194304, 41409225, 1073741824, 1329696225, 4294967296, 10667118605, 68719476736, 85530896451, 274877906944, 1371086188563, 17592186044416, 21972535073125, 70368744177664, 176021737014375

Offset: 0

Peter Luschny, Feb 11 2025

			r(n) = 0, 1, 1/2, 4/3, 9/16, 64/45, 75/128, 256/175, 1225/2048, ...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).

Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.

r := n -> (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2:
a := n -> numer(simplify(r(n))): seq(a(n), n = 0..28);
# Alternative:
r := n -> ifelse(n <= 1, n, (n - 1)/(n*r(n - 1))):

Join[{0}, Numerator[FoldList[(#2 - 1)/(#2*#) &, Range[30]]]] (* Paolo Xausa, Feb 14 2025 *)

Product_{k=1..n} a(k) = A380910(n) / A380909(n).
r(n) = (n - 1)/(n*r(n - 1)) for n > 1.
numerator(r(2*n)) = A161736(n).
numerator(r(2*n+1)) = A056982(n).
numerator(r(2*n+1))/4^n = A124399(n).
denominator(r(2*n-2)) = A161737(n).
denominator(r(2*n+1)) = A069955(n).
denominator(r(2*n+1))/(2*n+1) = A038534(n).
denominator(r(2*n+2))/2 = A278145(n).
denominator(r(2*n+2))/2^(2*n+1) = A001901(n).
r(n) ~ (2/Pi)^((-1)^n)*(1 - 1/(2*n) + 1/(8*n^2) + 1/(16*n^3) - 5/(128*n^4) - 23/(256*n^5) ...).

cp's OEIS Frontend

A380910 a(n) = denominator(n!! / (n - 1)!!).

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A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

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cp's OEIS Frontend

A380910 a(n) = denominator(n!! / (n - 1)!!).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.