A180000 -id:A180000 - cp's OEIS frontend

A195009 Triangle read by rows, T(n,k) = k^n*A056040(n), n>=0, 0<=k<=n.

1, 0, 1, 0, 2, 8, 0, 6, 48, 162, 0, 6, 96, 486, 1536, 0, 30, 960, 7290, 30720, 93750, 0, 20, 1280, 14580, 81920, 312500, 933120, 0, 140, 17920, 306180, 2293760, 10937500, 39191040, 115296020, 0, 70, 17920, 459270, 4587520, 27343750, 117573120, 403536070, 1174405120

Offset: 0

Peter Luschny, Sep 07 2011

			                     1
                    0, 1
                  0, 2, 8
               0, 6, 48, 162
            0, 6, 96, 486, 1536
       0, 30, 960, 7290, 30720, 93750
0, 20, 1280, 14580, 81920, 312500, 933120

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

swing := n -> n!/iquo(n,2)!^2: pow := (n,k) -> if k=0 and n=0 then 1 else n^k fi: A195009 := (n,k) -> pow(k,n)*swing(n):
# Formula:
omega := proc(x) BesselI(0,2*m*x)+(2*m*x+1)*BesselI(1,2*m*x) end:
f := n -> `if`(irem(n,2)=1,(n+1)/2,1/(n+1)): A195009 := proc(n,k)
limit(f(n)*(D@@n)(omega)(x),x=0); subs(m=k,%) end;

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; t[0, 0] = 1; t[n_, k_] := k^n*sf[n]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

T(n,k) = f(n)*lim(x=0, (d^n/dx)(BesselI(0,2*k*x)+(2*k*x+1) *BesselI(1,2*k*x) where f(n) = (n+1)/2 if n is odd, 1/(n+1) otherwise.

A163081 Primes of the form p$ + 1 where p is prime, where '$' denotes the swinging factorial (A056040).

Original entry on oeis.org

3, 7, 31, 4808643121, 483701705079089804581, 3283733939424401442167506310317720418331001

Offset: 1

Peter Luschny, Jul 21 2009

nonn

The values of p are 2, 3, 5, 31, 67, 139 which is A163079. Subsequence of A163075 (primes of the form k$ + 1).

			3 and 3$ + 1 = 7 are prime, so 7 is a member.

Jinyuan Wang, Table of n, a(n) for n = 1..7
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A056040, A103319, A163074-A163083.

a := proc(n) select(isprime,[$2..n]); select(isprime, map(x -> A056040(x)+1,%)) end:

A163082 Primes of the form p$ - 1 where p is prime, where '$' denotes the swinging factorial (A056040).

Original entry on oeis.org

5, 29, 139, 12011, 5651707681619, 386971244197199, 35257120210449712895193719, 815027488562171580969632861193966578650499

Offset: 1

Peter Luschny, Jul 21 2009

nonn

The first values of p are 3, 5, 7, 13, 41 from A163080. Subsequence of A163076 (primes of the form k$ - 1).

			3 and 3$ - 1 = 5 are prime, so 5 is a member.

Jinyuan Wang, Table of n, a(n) for n = 1..14
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A056040, A163074-A163083.

a := proc(n) select(isprime,[$2..n]); select(isprime, map(x -> A056040(x)-1,%)) end:

A163640 The radical of the swinging factorial A056040 for odd indices.

Original entry on oeis.org

1, 6, 30, 70, 210, 462, 6006, 4290, 72930, 461890, 1939938, 4056234, 6760390, 1560090, 6463230, 200360130, 2203961430, 907513530, 33578000610, 22974421470, 941951280270, 5786272150230, 526024740930, 1074920122770, 7524440859390, 25583098921926, 104300326374006, 1912172650190110

Offset: 0

Peter Luschny, Aug 02 2009

nonn

Let $ denote the swinging factorial. a(n) is the radical of (2*n+1)$ which is the product of the prime numbers dividing (2*n+1)$. It is the largest squarefree divisor of (2*n+1)$, and so also described as the squarefree kernel of (2*n+1)$.

			(2*5+1)$ = 2772 = 2^2*3^2*7*11. Therefore a(5) = 2*3*7*11 = 462.

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

A056040(n) = n$, A163641(n) = rad(n$), A080397(n) = rad((2n)$).

a := proc(n) local p; mul(p,p=numtheory[factorset]((2*n+1)!/iquo(2*n+1,2)!^2)) end:

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Times @@ FactorInteger[sf[2*n + 1]][[All, 1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 30 2013 *)

More terms from Michel Marcus, Aug 22 2025

A163644 Product of primes which do not exceed n and do not divide the swinging factorial n$ (A056040).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 5, 5, 35, 7, 7, 7, 21, 21, 105, 5, 55, 55, 165, 33, 429, 143, 1001, 1001, 1001, 1001, 1001, 91, 1547, 221, 221, 221, 4199, 323, 323, 323, 2261, 2261, 24871, 24871, 572033, 572033, 572033, 81719, 408595, 24035, 312455

Offset: 0

Peter Luschny, Aug 02 2009

nonn

			a(20) = 105 because in the prime-factorization of 20$ the primes 3, 5 and 7 are missing and 3*5*7 = 105.

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, A034386, A163641, A056610.

a := proc(n) local p; mul(p,p=select(isprime,{$1..n})
minus numtheory[factorset](n!/iquo(n,2)!^2)) end:

A034386[x_] := Apply[Times, Table[Prime[w], {w, 1, PrimePi[x]}]];
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f + 1, n - f]/f!];
A163641[0] = 1; A163641[n_] := Times @@ FactorInteger[sf[n]][[All, 1]]; Join[{1}, Table[A034386[n]/A163641[n], {n, 1, 50}]] (* G. C. Greubel, Aug 01 2017 *)

a(n) = primorial(n) / rad(n$) = A034386(n) / A163641(n).

A230714 a(n) = lcm(1,2,...,n)*a(floor(n/2))^2 for n > 0, a(0) = 1.

Original entry on oeis.org

1, 1, 2, 6, 48, 240, 2160, 15120, 1935360, 5806080, 145152000, 1596672000, 129330432000, 1681295616000, 82383485184000, 82383485184000, 2699542042509312000, 45892214722658304000, 413029932503924736000, 7847568717574569984000, 4904730448484106240000000

Offset: 0

Peter Luschny, Oct 28 2013

nonn

This sequence essentially documents the identity sqrt(A000142(n)*A205958(n)*A056040(n)*A180000(n)) = A003418(n)*a(floor(n/2)). Some interest derives from the fact that A056040 and A180000 can be computed by a structurally identical algorithm which can be used to compute A205958 and A000142, the latter being the fastest algorithm presently known to compute the factorial numbers. a(n) relates these numbers with lcm(1,2,3,...,n).

Michael De Vlieger, Table of n, a(n) for n = 0..351

Cf. A003418, A180000.

a := n -> `if`(n = 0, 1, lcm(seq(i,i=1..n))*a(floor(n/2))^2): seq(a(n),n=0..20);

Fold[Append[#1, Apply[LCM, Range@ #2] #1[[Floor[#2/2] + 1]]^2 ] &, {1}, Range@ 20] (* Michael De Vlieger, Mar 04 2018 *)

a(n)=lcm(vector(n,i,i))*if(n>3,a(n\2)^2,1) \\ Charles R Greathouse IV, Oct 31 2013

def A230714(n):
    return factorial(n)*A205958(n)
[A230714(n) for n in (0..20)]

a(n) = A205958(n)*n!.

A349269 Triangle read by rows, T(n, k) = (n - k)! * k! / floor(k / 2)! ^ 2.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 24, 6, 4, 6, 6, 120, 24, 12, 12, 6, 30, 720, 120, 48, 36, 12, 30, 20, 5040, 720, 240, 144, 36, 60, 20, 140, 40320, 5040, 1440, 720, 144, 180, 40, 140, 70, 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630

Offset: 0

Peter Luschny, Nov 13 2021

Interpolates between the factorial numbers (A000142) and the swinging factorial numbers (A056040).

The identity T(n, 0) = T(n, n)*T(floor(n/2), 0)^2 was investigated as a basis for an efficient implementation of the computation of the factorial numbers (see link).

			[0]      1;
[1]      1,     1;
[2]      2,     1,     2;
[3]      6,     2,     2,    6;
[4]     24,     6,     4,    6,   6;
[5]    120,    24,    12,   12,   6,  30;
[6]    720,   120,    48,   36,  12,  30,  20;
[7]   5040,   720,   240,  144,  36,  60,  20, 140;
[8]  40320,  5040,  1440,  720, 144, 180,  40, 140, 70;
[9] 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630;

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

Cf. A349270 (row sums), A193282 (central coeffs.), A000142, A056040, A180064.

T := (n, k) -> (n - k)!*k! / iquo(k,2)! ^ 2:
seq(seq(T(n, k), k = 0..n), n = 0..9);

T(n, k) divides T(n, 0) for 0 <= k <= n.

Product_{k=0..n} T(n, k) is a square.

A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457), read by rows.

Original entry on oeis.org

1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790

Offset: 0

Peter Luschny, Aug 07 2009

			Triangle begins:
   1;
  -1,   6;
   1, -12,   30;
  -1,  18,  -90,   140;
   1, -24,  180,  -560,   630;
  -1,  30, -300,  1400, -3150,   2772;
   1, -36,  450, -2800,  9450, -16632, 12012;

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

Row sums are the inverse binomial transform of the beta numbers (A163872).

Cf. A163649, A098473, A056040.

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, k) (-1)^(n-k)*binomial(n,k)*swing(2*k+1) end:
seq(print(seq(a(n,k),k=0..n)),n=0..8);

T[n_,k_] := ((-1)^(Mod[k,2]+n)*((2*k+1)!/(k!)^2)*Binomial[n,n-k]);
Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Oct 07 2023 *)

For n >= 0, k >= 0, T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ where i$ denotes the swinging factorial of i (A056040).
Conjectural g.f.: sqrt(1 + t)/(1 + (1 - 4*x)*t)^(3/2) = 1 + (-1 + 6*x)*t + (1 - 12*x + 30*x^2)*t^2 + .... - Peter Bala, Nov 10 2013
T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - Detlef Meya, Oct 07 2023

cp's OEIS Frontend

A195009 Triangle read by rows, T(n,k) = k^n*A056040(n), n>=0, 0<=k<=n.

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A163081 Primes of the form p$ + 1 where p is prime, where '$' denotes the swinging factorial (A056040).

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A163082 Primes of the form p$ - 1 where p is prime, where '$' denotes the swinging factorial (A056040).

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A163640 The radical of the swinging factorial A056040 for odd indices.

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A163644 Product of primes which do not exceed n and do not divide the swinging factorial n$ (A056040).

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A230714 a(n) = lcm(1,2,...,n)*a(floor(n/2))^2 for n > 0, a(0) = 1.

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A349269 Triangle read by rows, T(n, k) = (n - k)! * k! / floor(k / 2)! ^ 2.

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A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457), read by rows.

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