A004730 -id:A004730 - cp's OEIS frontend

A047037 T(n,n+2), array T as in A004730.

1, 3, 10, 35, 125, 455, 1680, 6270, 23591, 89335, 340076, 1300234, 4989504, 19206500, 74132230, 286802444, 1111864311, 4318284037, 16798674426, 65444081290, 255290775209, 997047334415, 3898217039280, 15256167628139

Offset: 0

Clark Kimberling

nonn

A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

Original entry on oeis.org

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

N. J. A. Sloane

Also numerator of rational part of Haar measure on Grassmannian space G(n,1).
Also rational part of numerator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A036039).
Let x(m) = x(m-2) + 1/x(m-1) for m >= 3, with x(1)=x(2)=1. Then the numerator of x(n+2) equals the denominator of n!!/(n+1)!! for n >= 0, where the double factorials are given by A006882. - Joseph E. Cooper III (easonrevant(AT)gmail.com), Nov 07 2010, as corrected in Cooper (2015).
Numerator of (n-1)/( (n-2)/( .../1)), with an empty fraction taken to be 1. - Flávio V. Fernandes, Jan 31 2025

			1, 1, (1/2)*Pi, 2, (3/4)*Pi, 8/3, (15/16)*Pi, 16/5, (35/32)*Pi, 128/35, (315/256)*Pi, ...
The sequence Gamma(n/2+1)/Gamma(n/2+1/2), n >= 0, begins 1/Pi^(1/2), 1/2*Pi^(1/2), 2/Pi^(1/2), 3/4*Pi^(1/2), 8/3/Pi^(1/2), 15/16*Pi^(1/2), 16/5/Pi^(1/2), ...

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.

T. D. Noe, Table of n, a(n) for n=0..302
Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.
Svante Janson, On the traveling fly problem, Graph Theory Notes of New York Vol. XXXI, 17, 1996.

Cf. A001803, A004730, A006882 (double factorials), A036069.

Cf. A036039, A046161, A001790, A001803, A101926.

import Data.Ratio ((%), numerator)
a004731 0 = 1
a004731 n = a004731_list !! n
a004731_list = map numerator ggs where
   ggs = 0 : 1 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
-- Reinhard Zumkeller, Dec 08 2011

if n mod 2 = 0 then k := n/2; 2*k*Pi*binomial(2*k-1,k)/4^k else k := (n-1)/2; 4^k/binomial(2*k,k); fi;
f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));
#
[1, seq(denom(doublefactorial(n-2)/doublefactorial(n-1)), n = 1..36)]; # Peter Luschny, Feb 09 2025

Table[ Denominator[ (n-2)!! / (n-1)!! ], {n, 0, 31}] (* Jean-François Alcover, Jul 16 2012 *)
Denominator[#[[1]]/#[[2]]&/@Partition[Range[-2,40]!!,2,1]] (* Harvey P. Dale, Nov 27 2014 *)
Join[{1},Table[Numerator[(n/2-1/2)!/((n/2-1)!Sqrt[Pi])], {n,1,31}]] (* Peter Luschny, Feb 08 2025 *)

f(n) = prod(i=0, (n-1)\2, n - 2*i); \\ A006882
a(n) = if (n==0, 1, denominator(f(n-2)/f(n-1))); \\ Michel Marcus, Feb 08 2025

from sympy import gcd, factorial2
def A004731(n):
    if n <= 1:
        return 1
    a, b = factorial2(n-2), factorial2(n-1)
    return b//gcd(a,b) # Chai Wah Wu, Apr 03 2021

Name corrected by Michel Marcus, Feb 08 2025

A076668 Decimal expansion of sqrt(2/Pi).

Original entry on oeis.org

7, 9, 7, 8, 8, 4, 5, 6, 0, 8, 0, 2, 8, 6, 5, 3, 5, 5, 8, 7, 9, 8, 9, 2, 1, 1, 9, 8, 6, 8, 7, 6, 3, 7, 3, 6, 9, 5, 1, 7, 1, 7, 2, 6, 2, 3, 2, 9, 8, 6, 9, 3, 1, 5, 3, 3, 1, 8, 5, 1, 6, 5, 9, 3, 4, 1, 3, 1, 5, 8, 5, 1, 7, 9, 8, 6, 0, 3, 6, 7, 7, 0, 0, 2, 5, 0, 4, 6, 6, 7, 8, 1, 4, 6, 1, 3, 8, 7, 2, 8, 6, 0, 6, 0

Offset: 0

Zak Seidov, Oct 25 2002

This is the limit of (n+1)!!/n!!/n^(1/2) at n_even->inf.

Expected value of |x - mu|/sigma for normal distribution with mean mu and standard deviation sigma (i.e., the normalized mean absolute deviation). - Stanislav Sykora, Jun 30 2017

			0.79788456080286535587989211986876373695171726232986931533...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Harmann König, Carsten Schütt, and Nicole Tomczak-Jaegermann, Projection constants of symmetric spaces and variants of Khintchine's inequality, J. reine angew. Math. 511 (1999), pp. 1-42.
Index entries for transcendental numbers

Cf. A004730, A004731, A019727, A060294 (Buffon's constant 2/Pi), A092678 (probable error).

pi:=Sqrt(2/Pi(RealField(110))); Reverse(Intseq(Floor(10^110*pi))); // Vincenzo Librandi, Jul 01 2017

RealDigits[Sqrt[2/Pi],10,120][[1]] (* Harvey P. Dale, Feb 05 2012 *)

sqrt(2/Pi) \\ G. C. Greubel, Sep 23 2017

Equals A087197*A002193. - R. J. Mathar Feb 05 2009

Equals integral_{-infinity..infinity} (1-erf(x)^2)/2 dx. - Jean-François Alcover, Feb 25 2015

More terms and better description from Benoit Cloitre and Michael Somos, Oct 29 2002

Leading zero removed, offset changed by R. J. Mathar, Feb 05 2009

A338719 Define b(1)=1 and for n>1, b(n)=n/b(n-1); then a(n) = ceiling(b(n)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 3, 5, 3, 5, 4, 6, 4, 6, 4, 6, 4, 6, 4, 7, 5, 7, 5, 7, 5, 7, 5, 8, 5, 8, 5, 8, 5, 8, 6, 8, 6, 9, 6, 9, 6, 9, 6, 9, 6, 9, 6, 10, 6, 10, 6, 10, 7, 10, 7, 10, 7, 10, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 8, 12, 8, 12, 8, 12, 8, 12, 8, 12, 8

Offset: 1

N. J. A. Sloane, Nov 29 2020, following a suggestion from Anchar Koops, Nov 24 2020

			The first few fractions b(n) are 1, 2, 3/2, 8/3, 15/8, 16/5, 35/16, 128/35, 315/128, 256/63, 693/256, 1024/231, 3003/1024, 2048/429, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A338718, A338720.

For the numerators and denominators of b(n) see A004731 and A004730.

R:= 1:
b:= 1:
for i from 2 to 100 do
  b:= i/b;
  R:= R, ceil(b)
od:
R; # Robert Israel, Jul 21 2024

A338720 Define b(1)=1 and for n>1, b(n)=n/b(n-1); then a(n) = nearest integer to b(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 2, 4, 3, 4, 3, 5, 3, 5, 3, 5, 4, 6, 4, 6, 4, 6, 4, 6, 4, 7, 4, 7, 4, 7, 5, 7, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 9, 5, 9, 6, 9, 6, 9, 6, 9, 6, 9, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 7, 10, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 7, 12, 7, 12, 7

Offset: 1

N. J. A. Sloane, Nov 29 2020, following a suggestion from Anchar Koops, Nov 24 2020

nonn

Since b(3) = 3/2, a(3) could also be taken to be 1.

			The first few fractions b(n) are 1, 2, 3/2, 8/3, 15/8, 16/5, 35/16, 128/35, 315/128, 256/63, 693/256, 1024/231, 3003/1024, 2048/429, ...

Cf. A338718, A338719.

For the numerators and denominators of b(n) see A004731 and A004730.

A338720b := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        n/procname(n-1) ;
    end if;
end proc:
A338720 := proc(n)
    round(A338720b(n)) ;
end proc:
seq(A338720(n),n=1..87) ; # R. J. Mathar, Dec 01 2020

b[n_] := b[n] = If[n == 1, 1, n/b[n-1]];
a[n_] := Round[b[n]];
Table[a[n], {n, 1, 87}] (* Jean-François Alcover, Apr 23 2023 *)

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

Peter Luschny, Feb 09 2025

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.

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

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

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);

A380909[n_] := Numerator[n!!/(n - 1)!!]; Array[A380909, 50, 0] (* or *)
Numerator[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).numerator for n in range(upto + 1)]
print(aList(35))

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

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

Original entry on oeis.org

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

A338718 Define b(1)=1 and for n>1, b(n)=n/b(n-1); then a(n) = floor(b(n)).

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 3, 2, 4, 2, 4, 2, 4, 3, 5, 3, 5, 3, 5, 3, 5, 3, 6, 4, 6, 4, 6, 4, 6, 4, 7, 4, 7, 4, 7, 4, 7, 5, 7, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 9, 5, 9, 5, 9, 6, 9, 6, 9, 6, 9, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11

Offset: 1

N. J. A. Sloane, Nov 29 2020, following a suggestion from Anchar Koops, Nov 24 2020

nonn

			The first few fractions b(n) are 1, 2, 3/2, 8/3, 15/8, 16/5, 35/16, 128/35, 315/128, 256/63, 693/256, 1024/231, 3003/1024, 2048/429, ...

Cf. A338719, A338720.

For the numerators and denominators of b(n) see A004731 and A004730.

a[n_] := Floor[n!!/(n - 1)!!]; Array[a, 90] (* Robert G. Wilson v, Dec 06 2020 *)

A086767 Last coefficient of the last term in the numerator of the simplified expansion of the solutions of FLT for n=2 for FLT n=1,2,3,..

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 1, 1, 1, 9, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 15, 1, 1, 1, 17, 1, 9, 1, 19, 1, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 25, 1, 13, 1, 27, 1, 7, 1, 29, 1, 15, 1, 31, 1, 1, 1, 33, 1, 17, 1, 35, 1, 9, 1, 37, 1, 19, 1, 39, 1, 5, 1, 41, 1, 21, 1, 43, 1, 11, 1, 45, 1, 23

Offset: 0

Cino Hilliard, Aug 02 2003

Integers a > b form the solution to FLT n = 2 as follows. (2ab)^2 = (a^2-b^2)^2 - (a^2+b^2)^2. The sequence is the coefficient c of the last b term in the numerator for the simplified expansion of the solution for n=2 as verification of FLT for n=1, 2, ...

			b/a
1
(3*a^4 + b^4)/(4*b*a^3)
(a^4 + b^4)/(2*b^2*a^2)
(5*a^8 + 10*b^4*a^4 + b^8)/(16*b^3*a^5)
(3*a^8 + 10*b^4*a^4 + 3*b^8)/(16*b^4*a^4)
(7*a^12 + 35*b^4*a^8 + 21*b^8*a^4 + b^12)/(64*b^5*a^7)
(a^12 + 7*b^4*a^8 + 7*b^8*a^4 + b^12)/(16*b^6*a^6)
(9*a^16 + 84*b^4*a^12 + 126*b^8*a^8 + 36*b^12*a^4 + b^16)/(256*b^7*a^9)
(5*a^16 + 60*b^4*a^12 + 126*b^8*a^8 + 60*b^12*a^4 + 5*b^16)/(256*b^8*a^8)
........
(K + cb^m)/2^m1b^m2c^m3
Seq = c for integers K,b,m1,m2,m3,n = 1,2,3...

Anonymous, Fermat's Theorem for Pythagorean Triples.

Cf. A004731, A004730.

sigma := proc(n) local i; add(i,i=convert(n,base,2)) end:
a := proc(n) if n=0 or type(n,odd) then 1 else if type(iquo(n,2),odd) then n/2 else n/2^(1-sigma(n)+sigma(n-1)) fi fi end: # Peter Luschny, Aug 03 2009

\ verification of general solution in integers \ a>b,x = 2ab,y=a^2-b^2,z=a^2+b^2 \ or FLT n=2 x^n+y^n <> z^n = (2ab)^n + (a^2-b^2)^n <> \(a^2+b^2)^n for n > 2 flt(n,a1,b1) = for(x=0,n,print(f(x,a1,b1))) f(n,a,b) = simplify(((a^2+b^2)^n - (a^2-b^2)^n)/(2*a*b)^n) coeffb(m) = { for(y=1,m, n=y; if(n%2,x=1, while(n%2==0,n=n/2); x=n; ); print1(x",") ) }

a(n) = A004731(n+1)/A004730(n). - Flávio V. Fernandes, Feb 13 2025

cp's OEIS Frontend

A047037 T(n,n+2), array T as in A004730.

Views

Author

Keywords

A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Extensions

A076668 Decimal expansion of sqrt(2/Pi).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A338719 Define b(1)=1 and for n>1, b(n)=n/b(n-1); then a(n) = ceiling(b(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A338720 Define b(1)=1 and for n>1, b(n)=n/b(n-1); then a(n) = nearest integer to b(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A338718 Define b(1)=1 and for n>1, b(n)=n/b(n-1); then a(n) = floor(b(n)).

Views

Author