A180000 -id:A180000 - cp's OEIS frontend

A163080 Primes p such that p$ - 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

3, 5, 7, 13, 41, 47, 83, 137, 151, 229, 317, 389, 1063, 2371, 6101, 7873, 13007, 19603

Offset: 1

Peter Luschny, Jul 21 2009

a(n) are the primes in A163078.

			3 is prime and 3$ - 1 = 5 is prime, so 3 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A056040, A103317, A163079, A163078.

a := proc(n) select(isprime,select(k -> isprime(A056040(k)-1),[$0..n])) end:

sf[n_] := n!/Quotient[n, 2]!^2; Select[Prime /@ Range[200], PrimeQ[sf[#] - 1] &] (* Jean-François Alcover, Jun 28 2013 *)

is(k) = isprime(k) && ispseudoprime(k!/(k\2)!^2-1); \\ Jinyuan Wang, Mar 22 2020

a(14)-a(18) from Jinyuan Wang, Mar 22 2020

A163211 Swinging Wilson quotients (A163210) which are primes.

Original entry on oeis.org

3, 23, 71, 757, 30671, 1383331, 245273927, 3362110459, 107752663194272623, 5117886516250502670227, 34633371587745726679416744736000996167729085703, 114326045625240879227044995173712991937709388241980425799

Offset: 1

Peter Luschny, Jul 24 2009

nonn

a(14)-a(18) certified prime by Primo 4.2.0. a(17) = A163210(569) = P1239, a(18) = A163210(787) = P1812. - Charles R Greathouse IV, Dec 11 2016

			The quotient (252+1)/11 = 23 is a swinging Wilson quotient and a prime, so 23 is a member.

G. C. Greubel, Table of n, a(n) for n = 1..16
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163210, A163213, A163212, A163209, A007619.

A163211 := n -> select(isprime,A163210(n));

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Select[PrimeQ][Table[a[n], {n, 1, 100}]] (* G. C. Greubel, Dec 10 2016 *)

sf(n)=n!/(n\2)!^2
forprime(p=2,1e3, t=sf(p-1)\/p; if(isprime(t), print1(t", "))) \\ Charles R Greathouse IV, Dec 11 2016

A163641 The radical of the swinging factorial A056040.

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 10, 70, 70, 210, 42, 462, 462, 6006, 858, 4290, 4290, 72930, 24310, 461890, 92378, 1939938, 176358, 4056234, 1352078, 6760390, 520030, 1560090, 222870, 6463230, 6463230, 200360130

Offset: 0

Peter Luschny, Aug 02 2009

nonn

The radical of n$ is the product of the prime numbers dividing n$. It is the largest squarefree divisor of n$, and so also described as the squarefree kernel of n$.

			11$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 2*3*7*11 = 462.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Bisections give: A080397 (even part), A163640 (odd part).

Cf. A056040.

a := proc(n) local p; mul(p,p=numtheory[factorset](n!/iquo(n,2)!^2)) end:

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Times @@ FactorInteger[sf[n]][[All, 1]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *)

a(n) = rad(n$).

A163774 Row sums of the central coefficients triangle (A163771).

Original entry on oeis.org

1, 3, 13, 51, 201, 783, 3039, 11763, 45481, 175803, 679779, 2630367, 10187659, 39500373, 153329913, 595883763, 2318471289, 9030982491, 35216266947, 137469149451, 537152523711, 2100857828193, 8223917499477, 32219655346719, 126328429601451, 495676719721953, 1946227355491909

Offset: 0

Peter Luschny, Aug 05 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A056040, A163771.

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i,k; add(add((-1)^(n-i)*binomial(n-k,n-i)*swing(2*i),i=k..n), k=0..n) end:

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$, where i$ denotes the swinging factorial of i (A056040).

Conjecture: a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+1,k)*binomial(2*k,k). - Werner Schulte, Nov 17 2015

More terms from Michel Marcus, Nov 24 2015

A163865 The binomial transform of the swinging factorial (A056040).

Original entry on oeis.org

1, 2, 5, 16, 47, 146, 447, 1380, 4251, 13102, 40343, 124136, 381625, 1172198, 3597401, 11031012, 33798339, 103477590, 316581567, 967900224, 2957316429, 9030317478, 27558851565, 84059345244, 256265811333, 780885245826, 2378410969977, 7241027262280

Offset: 0

Peter Luschny, Aug 06 2009

nonn

a(n) = Sum_{k=0..n} binomial(n,k) * k$, where k$ denotes the swinging factorial of k (A056040). The swinging analog to the number of arrangements, the binomial transform of the factorial (A000522).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A056040, A000522, A163650.

a := proc(n) local k: add(binomial(n,k)*(k!/iquo(k, 2)!^2),k=0..n) end:
seq(coeff(series((1-z-4*z^2)/((1+z)*(1-3*z))^(3/2),z,28),z,n),n=0..27); # Peter Luschny, Oct 31 2013

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Sum[Binomial[n, k]*sf[k], {k, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 26 2013 *)
sf[n_] := n!/Quotient[n, 2]!^2; t[n_] := Sum[Binomial[n, k]*sf[k], {k, 0, n}]; Table[t[n], {n,0,50}] (* G. C. Greubel, Aug 06 2017 *)

x='x+O('x^50); Vec((1-x-4*x^2)/((1+x)*(1-3*x))^(3/2)) \\ G. C. Greubel, Aug 06 2017

E.g.f.: exp(x)*BesselI(0,2*x)*(1+x). - Peter Luschny, Aug 26 2012
O.g.f.: (1-x-4*x^2)/((1+x)*(1-3*x))^(3/2). - Peter Luschny, Oct 31 2013
a(n) ~ 3^(n - 1/2) * sqrt(n) / (2*sqrt(Pi)). - Vaclav Kotesovec, Nov 27 2017

A189230 Complementary Catalan triangle read by rows.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 0, 8, 0, 4, 0, 10, 0, 15, 0, 5, 0, 0, 30, 0, 24, 0, 6, 0, 35, 0, 63, 0, 35, 0, 7, 0, 0, 112, 0, 112, 0, 48, 0, 8, 0, 126, 0, 252, 0, 180, 0, 63, 0, 9, 0, 0, 420, 0, 480, 0, 270, 0, 80, 0, 10, 0, 462, 0, 990, 0, 825, 0, 385, 0, 99, 0, 11, 0

Offset: 0

Peter Luschny, May 01 2011

T(n,k) = A189231(n,k)*((n - k) mod 2). For comparison: the classical Catalan triangle is A053121(n,k) = A189231(n,k)*((n-k+1) mod 2).

T(n,0) = A138364(n). Row sums: A100071.

			[0]  0,
[1]  1,  0,
[2]  0,  2,  0,
[3]  3,  0,  3,  0,
[4]  0,  8,  0,  4,  0,
[5] 10,  0, 15,  0,  5, 0,
[6]  0, 30,  0, 24,  0, 6, 0,
[7] 35,  0, 63,  0, 35, 0, 7, 0,
   [0],[1],[2],[3],[4],[5],[6],[7]

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers

Cf. A053121, A162246, A057977, A189231.

A189230 := (n,k) -> A189231(n,k)*modp(n-k,2):
seq(print(seq(A189230(n,k),k=0..n)),n=0..11);

t[n_, k_] /; (k>n || k<0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2] t[n-1, k] + t[n-1, k+1];
T[n_, k_] := t[n, k] Mod[n-k, 2];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jun 24 2019 *)

A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

Original entry on oeis.org

1, 6, 5, 28, 9, 22, 13, 120, 17, 38, 21, 92, 25, 54, 29, 496, 33, 70, 37, 156, 41, 86, 45, 376, 49, 102, 53, 220, 57, 118, 61, 2016, 65, 134, 69, 284, 73, 150, 77, 632, 81, 166, 85, 348, 89, 182, 93, 1520, 97, 198, 101, 412, 105, 214, 109, 888, 113, 230, 117

Offset: 1

Peter Luschny, Dec 01 2012

An asymptotic expansion of the logarithm of the central binomial (A000984) for n>0 is given by log(binomial(2*n,n)) ~ (n*log(16)-log(Pi)-log(n) + sum_{k>=1}((-4)^(-k)*A002425(k)/a(k)*n^(1-2*k)))/2.

An asymptotic expansion of the logarithm of the swinging factorial (A056040) for n>1 is given by log(swing(n)) ~ (n*log(4)-log(Pi)-(-1)^n*(log(n/2) - (1/2)*sum_{k>=1}((-1)^k*A002425(k)/a(k)*n^(1-2*k))))/2.

			log(binomial(2*n,n)) = n*log(4) - (log(n)+log(Pi))/2 - 1/(8*a(1)*n) + 1/(32*a(2)*n^3) - 1/(128*a(3)*n^5) + 17/(512*a(4)*n^7) - 31/(2048*a(5)*n^9) + 691/(8192*a(6)*n^11) + O(1/n^13).
log(swing(n)) = n*log(2) - (1/2)*log(Pi) - (1/4)*(-1)^n*(2*log(n/2) + 1/(a(1)*n) - 1/(a(2)*n^3) + 1/(a(3)*n^5) - 17/(a(4)*n^7) + 31/(a(5)*n^9) - 691/(a(6)*n^11)) + O(1/n^13).

Peter Luschny, Table of n, a(n) for n = 1..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A006519, A118413, A385054.

Coeff_list := proc(len) local n;
asympt(ln(n/2)/2+lnGAMMA(n/2+1/2)-lnGAMMA(n/2+1),n,2*len+3);
subs(n=1/n,simplify(convert(%,polynom)));
[seq(4*coeff(unapply(%,n)(n),n,2*k+1),k=0..len-1)] end:
A219931_list := n -> denom(Coeff_list(n)); A219931_list(59);

max = 60; s = Normal[Series[Log[x/2]/2+LogGamma[x/2+1/2]-LogGamma[x/2+1], {x, Infinity, 2*max}]] /. x -> 1/x; a[n_] := Denominator[4*Coefficient[s, x^(2*n-1), 1]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Feb 17 2014 *)
a[n_] := Denominator[2*EulerE[2*n-1, 1]/(2*n-1)]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Apr 04 2014, after Peter Luschny *)

a(n) = denominator(2*E(2*n-1, 1)/(2*n-1)) where E(n, x) is the Euler polynomial. - Peter Luschny, Apr 03 2014

Warning: a(n) != (2*n-1)*2^valuation(n, 2). This was mistakenly assumed several times in the past, see A385054. - Peter Luschny, Jun 17 2025

Edited and incorrect entries removed by Georg Fischer and Peter Luschny, Jun 16 2025

A263673 a(n) = lcm{1,2,...,n} / binomial(n,floor(n/2)).

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 3, 12, 12, 20, 10, 60, 30, 210, 105, 56, 56, 504, 252, 2520, 1260, 660, 330, 3960, 1980, 5148, 2574, 4004, 2002, 30030, 15015, 240240, 240240, 123760, 61880, 31824, 15912, 302328, 151164, 77520, 38760, 813960, 406980, 8953560, 4476780, 2288132, 1144066, 27457584, 13728792, 49031400

Offset: 0

Max Alekseyev, Oct 23 2015

From Robert Israel, Oct 23 2015: (Start)
If n = 2^k, a(n) = a(n-1).
If n = p^k where p is an odd prime and k >= 1, 2*n*a(n) = p*(n+1)*a(n-1).
If n is even and not a prime power, 2*a(n) = a(n-1).
If n is odd and not a prime power, 2*n*a(n) = (n+1)*a(n-1). (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A068550, A180000.

a := n -> lcm(seq(k,k=1..n))/binomial(n,iquo(n,2)):
seq(a(n), n=0..49); # Peter Luschny, Oct 23 2015

Join[{1}, Table[LCM @@ Range[n]/Binomial[n, Floor[n/2]], {n, 1, 50}]] (* or *) Table[Product[Cyclotomic[k, 1], {k, 2, n}]/Binomial[n, Floor[n/2]], {n, 0, 50}] (* G. C. Greubel, Apr 17 2017 *)

A263673(n) = lcm(vector(n,i,i)) / binomial(n,n\2);

a(n) = A003418(n) / A001405(n).
a(n) = A048619(n-1) * A110654(n).
a(2*n) = A068550(n) = A099996(n) / A000984(n).
a(n) = A180000(n)*A152271(n). - Peter Luschny, Oct 23 2015
a(n) = (e/2)^(n + o(1)). - Charles R Greathouse IV, Oct 23 2015

A163775 Row sums of triangle A163772.

Original entry on oeis.org

1, 11, 73, 403, 2021, 9567, 43611, 193683, 844213, 3629083, 15437951, 65143503, 273148279, 1139548469, 4734740493, 19606960755, 80969809797, 333601494651

Offset: 0

Peter Luschny, Aug 05 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A163772.

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i,k; add(add((-1)^(n-i)*binomial(n-k,n-i)*swing(2*i+1),i=k..n),k=0..n) end:

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)* Binomial[n - k, n - i]*sf[2*i + 1], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).

A182918 Denominators of the swinging Bernoulli number b_n.

Original entry on oeis.org

1, 2, 6, 1, 120, 1, 1512, 1, 17280, 1, 190080, 1, 1415232000, 1, 21772800, 1, 829108224000, 1, 105082151731200, 1, 4345502515200000, 1, 19989311569920000, 1, 626378114550988800000, 1, 17896517558599680000, 1, 944578196742891110400000

Offset: 0

Peter Luschny, Feb 03 2011

Let zeta(n) denote the Riemann zeta function, B_n the Bernoulli numbers and let [n even] be 1 if n is even, 0 otherwise.
Then 2 zeta(n) [n even] = (2 Pi)^n | B_n | / n! for n >= 2.
Replacing in this formula the factorial of n by the swinging factorial of n (A056040) defines the 'swinging Bernoulli number' b_n.
Then 2 zeta(n) [n even] = (2 Pi)^n b_n / n$ for n >= 2.
Let additionally b_0 = 1 and b_1 = 1/2. The b_n are rational numbers like the Bernoulli numbers; unlike the Bernoulli numbers the swinging Bernoulli numbers are unsigned, bounded in the interval [0,1] and approach 0 for n -> infinity. The numerators of the swinging Bernoulli numbers b_n are abs(A120082(n)).

			1, 1/2, 1/6, 0, 1/120, 0, 1/1512, 0, 1/17280, 0, 1/190080, ..

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A120082.

swbern:= proc(n) local swfact;
swfact := n -> n!/iquo(n,2)!^2;
if n=0 then 1 elif n=1 then 1/2 else
   if n mod 2 = 1 then 0
   else 2*Zeta(n)*swfact(n)/(2*Pi)^n fi
fi end:
Abs_A120082 := n -> numer(swbern(n));
A182918 := n -> denom(swbern(n));
seq(A182918(i),i=0..20);

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[1] = 2; a[?OddQ] = 1; a[n] := 2*Zeta[n]*sf[n]/(2*Pi)^n // Denominator; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jul 26 2013 *)

cp's OEIS Frontend

A163080 Primes p such that p$ - 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A163211 Swinging Wilson quotients (A163210) which are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A163641 The radical of the swinging factorial A056040.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A163774 Row sums of the central coefficients triangle (A163771).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A163865 The binomial transform of the swinging factorial (A056040).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A189230 Complementary Catalan triangle read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs