cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Peter Luschny, Feb 09 2025

Keywords

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

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

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

Views

Author

Peter Luschny, Feb 11 2025

Keywords

Examples

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

Links

Crossrefs

Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).
Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.

Programs

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

Formula

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) ...).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.