A014973 -id:A014973 - cp's OEIS frontend

A049077 a(n) = n / gcd(n, binomial(n, floor(n/2))).

1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 1, 1, 7, 1, 8, 1, 9, 1, 5, 1, 11, 1, 6, 1, 13, 1, 7, 1, 1, 1, 16, 1, 17, 1, 3, 1, 19, 1, 2, 1, 7, 1, 11, 1, 23, 1, 4, 1, 25, 1, 13, 1, 27, 1, 1, 1, 29, 1, 15, 1, 31, 1, 32, 1, 11, 1, 17, 1, 5, 1, 18, 1, 37, 1, 19, 1, 39, 1, 4, 1, 41, 1, 1, 1, 43, 1, 11, 1, 1, 1, 23

Offset: 1

Labos Elemer, Nov 13 2000

nonn

			For n = 12, gcd(12, binomial(12, 6)) = gcd(12, 924) = 12, so a(12) = 1.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001405, A057815, A014973.

swing := n -> n!/iquo(n,2)!^2: seq(n/igcd(n,swing(n)),n=1..92); # Peter Luschny, May 16 2013

Flatten[Table[{1, n/GCD[n, Binomial[n, n/2]]}, {n, 2, 100, 2}]] (* Alonso del Arte, May 17 2013 *)

a(n) = n/gcd(n, binomial(n, n\2)); \\ Michel Marcus, Mar 22 2020

For odd n, a(n) = 1. For even n, a(n) is either n/2 or smaller.

Offset changed to 1 by Peter Luschny, May 16 2013

A092043 a(n) = numerator(n!/n^2).

Original entry on oeis.org

1, 1, 2, 3, 24, 20, 720, 630, 4480, 36288, 3628800, 3326400, 479001600, 444787200, 5811886080, 81729648000, 20922789888000, 19760412672000, 6402373705728000, 6082255020441600, 115852476579840000, 2322315553259520000

Offset: 1

Ralf Stephan, Mar 28 2004

Numerator of expansion of dilog(x) = Li_2(x) = -Integral_{t=0..x} (log(1-t)/t)*dt. See the Weisstein link.

E.g.f. of {a(n)/A014973(n)}_{n>=1} is Li_2(x) (with 0 for n=0).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
A. N. Kirillov, Dilogarithm identities, arXiv:hep-th/9408113, 1994.
Eric Weisstein's World of Mathematics, Dilogarithm

Denominator is in A014973.

Cf. A000142, A001819, A181569.

[Numerator(Factorial(n)/n^2): n in [1..30]]; // Vincenzo Librandi, Apr 15 2014

Table[Numerator[n!/n^2], {n, 1, 40}] (* Vincenzo Librandi, Apr 15 2014 *)
Table[(n-1)!/n,{n,30}]//Numerator (* Harvey P. Dale, Apr 03 2018 *)

PARI
```
a(n)=numerator(n!/n^2)
```

a(n)=numerator(polcoeff(serlaplace(dilog(x)),n))

From Wolfdieter Lang, Apr 28 2017: (Start)
a(n) = numerator(n!/n^2) = numerator((n-1)!/n), n >= 1. See the name.
E.g.f. {a(n)/A014973(n)}_{n>=1} with 0 for n=0 is  Li_2(x). See the comment.
(-1)^n*a(n+1)/A014973(n+1) = (-1)^n*n!/(n+1) = Sum_{k=0..n} Stirling1(n, k)*Bernoulli(k), with Stirling1 = A048994 and Bernoulli(k) = A027641(k)/A027642(k), n >= 0. From inverting the formula for B(k) in terms of Stirling2 = A048993.(End)
From Wolfdieter Lang, Oct 26 2022: (Start)
a(n) = (n-1)!/gcd(n,(n-1)!) = A000142(n-1)/A181569(n-1), n >= 1.
The expansion of (1+x)*exp(x) has coefficients A014973(n+1)/a(n+1), for n >= 0. (End)

Comment rewritten by Wolfdieter Lang, Apr 28 2017

A181569 Greatest common divisor of n! and n+1.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 64, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 75, 76, 77, 78, 1

Offset: 1

Reinhard Zumkeller, Oct 31 2010

From Wilson's theorem, it follows that a(n) = 1 when n + 1 is prime, a(n) > 1 otherwise. - Alonso del Arte, Feb 25 2014

			a(6) = 1 because 6! and 7 are coprime.
a(7) = 8 because 7! = 5040 and gcd(5040, 8) = 8.
a(8) = 9 because 8! = 40320 and gcd(40320, 9) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000142, A006093, A010051, A050873.

Cf. A088140, A135683.

[GCD(Factorial(n), n+1): n in [1..80]]; // Vincenzo Librandi, Mar 03 2014

A181569:=n->gcd(n!,n+1): seq(A181569(n), n=1..100); # Wesley Ivan Hurt, Aug 13 2015

Table[GCD[n!, n + 1], {n, 80}] (* Alonso del Arte, Feb 25 2014 *)

a(n)= n!/denominator(polcoeff((x+1)*exp(x+x*O(x^n)), n));  \\ Gerry Martens, Aug 12 2015

A181569(n)=gcd(n!,n+1) \\ M. F. Hasler, Aug 16 2015

a(n) = A050873(A000142(n), n + 1);
a(A006093(n)) = 1;
for n > 3: a(n) = (n + 1) / (n*A010051(n+1) + 1).
a(n) = (n+1)/A014973(n+1). - Michel Marcus, Aug 14 2015

A343206 Numerators of Daehee numbers.

Original entry on oeis.org

1, -1, 2, -3, 24, -20, 720, -630, 4480, -36288, 3628800, -3326400, 479001600, -444787200, 5811886080, -81729648000, 20922789888000, -19760412672000, 6402373705728000, -6082255020441600, 115852476579840000, -2322315553259520000, 1124000727777607680000, -1077167364120207360000

Offset: 0

Michel Marcus, Apr 08 2021

			1, -1/2, 2/3, -3/2, 24/5, -20, 720/7, -630, 4480, -36288, 3628800/11, -3326400, 479001600/13, -444787200, ...

Dae San Kim and Taekyun Kim, Daehee Numbers and polynomials, arXiv:1309.2109 [math.NT], 2013.
Dae San Kim and Taekyun Kim, Daehee numbers and polynomials, Applied Mathematical Sciences, Vol. 7, 2013, no. 120, 5969-5976.

Cf. A008275 (Stirling1), A027641/A027642 (Bernoulli).

Cf. A014973 (denominators).

a[n_]:=Numerator[(-1)^n*n!/(n+1)]; Array[a,24,0] (* Stefano Spezia, Jun 24 2024 *)

a(n) = numerator(sum(i=0, n, stirling(n, i, 1)*bernfrac(i)));

my(x='x+O('x^30), v=Vec(serlaplace(log(1+x)/x))); apply(numerator,v)

from sympy.functions.combinatorial.numbers import stirling, bernoulli
def A343206(n): return sum(stirling(n,i,signed=True)*bernoulli(i) for i in range(n+1)).p # Chai Wah Wu, Apr 08 2021

D(n) = Sum_{i=0..n} Stirling1(n, i)*Bernoulli(i).
E.g.f. for D(n): log(1+x)/x.
D(n) = a(n)/A014973(n+1).
a(n) = numerator((-1)^n*n!/(n+1)). - Stefano Spezia, Jun 24 2024

A181857 a(n) = lcm(n^2, n!).

Original entry on oeis.org

0, 1, 4, 18, 48, 600, 720, 35280, 40320, 362880, 3628800, 439084800, 479001600, 80951270400, 87178291200, 1307674368000, 20922789888000, 6046686277632000, 6402373705728000, 2311256907767808000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 0

Peter Luschny, Nov 21 2010

nonn

If n > 4 then a(n) = n! for composite n and n * n! for prime n. - David A. Corneth, Aug 04 2015

Robert Israel, Table of n, a(n) for n = 0..411

Cf. A170826, A181860, A056040, A000290, A014973.

[Lcm(n^2, Factorial(n)): n in [0..30]]; // Vincenzo Librandi, Aug 07 2015

Maple
```
A181857 := n -> ilcm(n^2, n!);
```

Table[LCM[n^2, n!], {n, 0, 17}] (* Michael De Vlieger, Aug 04 2015 *)

a(n)=lcm(n!,n^2) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = lcm(A000290(n), A000142(n)).

a(n) = n! * A014973(n) for n >= 1. - Johannes W. Meijer, Jun 04 2016

A160039 Numerators of n!*(1 + 1/2 + 1/3 +...+ 1/(n+1)).

Original entry on oeis.org

1, 3, 11, 25, 274, 294, 13068, 13698, 114064, 1062864, 120543840, 123870240, 19802759040, 20247546240, 289277533440, 4420892649600, 1223405590579200, 1243166003251200, 431565146817638400, 437647401838080000

Offset: 0

Peter Luschny, Apr 30 2009

The denominators of n!*(1 + 1/2 + 1/3 +...+ 1/(n+1)) are A014973 with offset 0.

Cf. A014973 (denominators).

FH := n -> numer(n!*add(1/i,i=1..n+1)); seq(FH(i),i=0..20);

Table[Numerator[n! HarmonicNumber[n+1]], {n, 0, 19}] (* Jean-François Alcover, Jun 17 2019 *)

A240533 a(n) = numerators of n!/10^n.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 9, 63, 63, 567, 567, 6237, 18711, 243243, 1702701, 5108103, 5108103, 86837751, 781539759, 14849255421, 14849255421, 311834363841, 3430178002251, 78894094051773, 236682282155319, 236682282155319, 3076869668019147, 83075481036516969

Offset: 0

Vincenzo Librandi, Apr 14 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000142, A011557, A014973, A074143, A092043, A208529.

[Numerator(Factorial(n)/10^n): n in [0..30]];

Mathematica
```
Table[Numerator[n!/10^n], {n, 0, 30}]
```

A240534 a(n) = denominators of n!/10^n.

Original entry on oeis.org

1, 10, 50, 500, 1250, 2500, 12500, 125000, 156250, 1562500, 1562500, 15625000, 39062500, 390625000, 1953125000, 3906250000, 2441406250, 24414062500, 122070312500, 1220703125000, 610351562500, 6103515625000, 30517578125000

Offset: 0

Vincenzo Librandi, Apr 14 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000142, A011557, A014973, A074143, A092043, A208529.

[Denominator(Factorial(n)/10^n): n in [0..30]];

Mathematica
```
Table[Denominator[n!/10^n], {n, 0, 30}]
```

A344849 a(n) is the numerator of Catalan-Daehee number d(n).

Original entry on oeis.org

1, 1, 7, 20, 313, 344, 24634, 86008, 183349, 3301264, 132174038, 69326344, 3332927794, 17361255440, 108222173516, 406589577424, 26070625295573, 8970328188896, 55462481190898, 1055714050810664, 2169454884422962, 91277283963562352, 8046203518285051612, 686567135431420560

Offset: 0

Stefano Spezia, May 30 2021

Dae San Kim and Taekyun Kim, A new approach to Catalan numbers using differential equations, Russ. J. Math. Phys. 24, 465-475 (2017).
Taekyun Kim and Dae San Kim, Some identities of Catalan-Daehee polynomials arising from umbral calculus, Appl. Comput. Math. 16 (2017), no. 2, 177-189.
Yuankui Ma, Taekyun Kim, Dae San Kim and Hyunseok Lee, Study on q-analogues of Catalan-Daehee numbers and polynomials, arXiv:2105.12013 [math.NT], 2021.

Cf. A000108, A000302, A014973 (denominators of Daehee numbers), A343206, A344850 (denominators).

nmax:=24; a[n_]:=Numerator[Coefficient[Series[Log[1-4x]/(2(Sqrt[1-4x]-1)),{x,0,nmax}],x,n]]; Array[a,nmax,0] (* or *)
a[n_]:=Numerator[If[n==0,1,4^n/(n+1)-Sum[4^(n-m-1)CatalanNumber[m]/(n-m),{m,0,n-1}]]]; Array[a,24,0]

G.f. of d(n): log(1 - 4*x)/(2*(sqrt(1 - 4*x) - 1)).

a(n) = numerator(d(n)), where d(n) = 4^n/(n + 1) - Sum_{m=0..n-1} 4^(n-m-1)*C(m)/(n - m) with d(0) = 1 and C(m) is the m-th Catalan number.

A344850 a(n) is the denominator of Catalan-Daehee number d(n).

Original entry on oeis.org

1, 1, 3, 3, 15, 5, 105, 105, 63, 315, 3465, 495, 6435, 9009, 15015, 15015, 255255, 23205, 37791, 188955, 101745, 1119195, 25741485, 572033, 42902475, 79676025, 42181425, 42181425, 155687805, 40970475, 1270084725, 1270084725, 665282475, 173996955, 6089893425, 794333925

Offset: 0

Stefano Spezia, May 30 2021

Dae San Kim and Taekyun Kim, A new approach to Catalan numbers using differential equations, Russ. J. Math. Phys. 24, 465-475 (2017).
Taekyun Kim and Dae San Kim, Some identities of Catalan-Daehee polynomials arising from umbral calculus, Appl. Comput. Math. 16 (2017), no. 2, 177-189.
Yuankui Ma, Taekyun Kim, Dae San Kim and Hyunseok Lee, Study on q-analogues of Catalan-Daehee numbers and polynomials, arXiv:2105.12013 [math.NT], 2021.

Cf. A000108, A000302, A014973 (denominators of Daehee numbers), A343206, A344849 (numerators).

nmax:=36; a[n_]:=Denominator[Coefficient[Series[Log[1-4x]/(2(Sqrt[1-4x]-1)),{x,0,nmax}],x,n]]; Array[a,nmax,0] (* or *)
a[n_]:=Denominator[If[n==0,1,4^n/(n+1)-Sum[4^(n-m-1)CatalanNumber[m]/(n-m),{m,0,n-1}]]]; Array[a,36,0]

G.f. of d(n): log(1 - 4*x)/(2*(sqrt(1 - 4*x) - 1)).

a(n) = denominator(d(n)), where d(n) = 4^n/(n + 1) - Sum_{m=0..n-1} 4^(n-m-1)*C(m)/(n - m) with d(0) = 1 and C(m) the m-th Catalan number.

cp's OEIS Frontend

A049077 a(n) = n / gcd(n, binomial(n, floor(n/2))).

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A092043 a(n) = numerator(n!/n^2).

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A181569 Greatest common divisor of n! and n+1.

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A343206 Numerators of Daehee numbers.

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A181857 a(n) = lcm(n^2, n!).

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A160039 Numerators of n!*(1 + 1/2 + 1/3 +...+ 1/(n+1)).

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A240533 a(n) = numerators of n!/10^n.

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