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a(n) = lcm{1,2,...,n} / swinging_factorial(n) = A003418(n) / A056040(n).
+20
49
1, 1, 1, 1, 2, 2, 3, 3, 12, 4, 10, 10, 30, 30, 105, 7, 56, 56, 252, 252, 1260, 60, 330, 330, 1980, 396, 2574, 286, 2002, 2002, 15015, 15015, 240240, 7280, 61880, 1768, 15912, 15912, 151164, 3876, 38760, 38760, 406980, 406980, 4476780, 99484, 1144066
COMMENTS
Characterization: Let e_{p}(m) denote the exponent of the prime p in the prime factorization of m and [.] denote the Iverson bracket, then
e_{p}(a(n)) = Sum_{k>=1} [floor(n/p^k) is even].
This implies, among other things, that no prime > floor(n/2) can divide a(n). The prime exponents e_{2}(a(2n)) give Guy Steele's sequence GS(5,3) A080100. Asymptotics: log a(n) ~ n(1 - log 2). It is conjectured that log a(n) ~ n(1 - log 2) + O(n^{1/2+eps}) for all eps > 0.
Bounds: A056040(floor(n/3)) <= a(n) <= A056040(floor(n/2)) if n >= 285.
FORMULA
a(n) = 2^(-n)*Product_{1<=k<=n} A014963(k)*(k/2)^((-1)^k).
MAPLE
A014963 := proc(n) if n < 2 then 1 else numtheory[factorset](n);
if 1 < nops(%) then 1 else op(%) fi fi end;
mul( A014963(k)*(k/2)^((-1)^k), k=1..n)/2^n end; # Also:
lcm := ilcm(seq(i, i=1..n));
sf := n!/iquo(n, 2)!^2;
lcm/sf end;
MATHEMATICA
a[0] = 1; a[n_] := LCM @@ Range[n] / (n! / Floor[n/2]!^2); Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Jul 23 2013 *)
PROG
(PARI) L=1; X(n)={ ispower(n, , &n); if(isprime(n), n, 1); }
Y(n)={ a=X(n); b=if(bitand(1, n), a, a*(n/2)^2); L=(b*L)/n; }
A180000_list(n)={ L=1; vector(n, m, Y(m)); } \\ for n>0
(Sage)
def Exp(m, n) :
s = 0; p = m; q = n//p
while q > 0 :
if is_even(q) :
s = s + 1
p = p * m
q = n//p
return s
A = [1, 1, 1, 1, 2, 2, 3, 3, 12]
if n < 9 : return A[n]
R = []; r = isqrt(n)
P = Primes(); p = P.first()
while p <= n//2 :
if p <= r : R.append(p^Exp(p, n))
elif p <= n//3 :
if is_even(n//p) : R.append(p)
else : R.append(p)
p = P.next(p)
return mul(x for x in R)
Product of first n swinging factorials ( A056040).
+20
9
1, 1, 2, 12, 72, 2160, 43200, 6048000, 423360000, 266716800000, 67212633600000, 186313420339200000, 172153600393420800000, 2067909047925770649600000, 7097063852481244869427200000
COMMENTS
With the definition of the Hankel transform as given by Luschny (see link) which uniquely determines the original sequence (provided that all determinants are not zero) this is also 1/ the Hankel determinant of 1/(n+1) (assuming (0,0)-based matrices).
a(2*n-1) is 1/determinant of the Hilbert matrix H(n) ( A005249).
MAPLE
a := proc(n) local i; mul( A056040(i), i=0..n) end;
MATHEMATICA
a[0] = 1; a[n_] := a[n] = a[n-1]*n!/Floor[n/2]!^2; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 26 2013 *)
PROG
(Sage)
swing = lambda n: factorial(n)/factorial(n//2)^2
return mul(swing(i) for i in (0..n))
Subswing - the inverse binomial transform of the swinging factorial ( A056040).
+20
9
1, 0, 1, 2, -9, 44, -165, 594, -2037, 6824, -22437, 72830, -234047, 746316, -2364947, 7455798, -23405085, 73207728, -228275949, 709906518, -2202557691, 6819616020, -21076580511, 65032888998, -200369138571, 616531573224, -1894784517675, 5816886949874
COMMENTS
Analog to the subfactorial A000166.
FORMULA
E.g.f.: exp(-x)*BesselI(0,2*x)*(1+x). - Peter Luschny, Aug 26 2012 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)*(k!/(floor(k/2)!)^2). - G. C. Greubel, Aug 01 2017 a(n) ~ -(-1)^n * sqrt(n) * 3^(n - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Oct 31 2017 D-finite with recurrence n*a(n) +5*(n-1)*a(n-1) +(n-4)*a(n-2) +(-13*n+23)*a(n-3) +6*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 04 2023
MAPLE
a := proc(n) local k: add((-1)^(n-k)*binomial(n, k)*(k!/iquo(k, 2)!^2), k=0..n) end:
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*sf[k], {k, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 28 2013 *)
PROG
(PARI) for(n=0, 50, print1(sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(k!/((k\2)!)^2)), ", ")) \\ G. C. Greubel, Aug 01 2017
Run Length Transform of swinging factorials ( A056040).
+20
9
1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 2, 2, 6, 6, 1, 1, 1, 2, 1, 1, 2, 6, 2, 2, 2, 4, 6, 6, 6, 30, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 2, 2, 6, 6, 2, 2, 2, 4, 2, 2, 4, 12, 6, 6, 6, 12, 6, 6, 30, 20, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 2, 2, 6, 6, 1, 1, 1, 2, 1, 1
COMMENTS
For the definition of the Run Length Transform see A246595.
FORMULA
a(2^n-1) = n$ where n$ is the swinging factorial of n, A056040(n).
MATHEMATICA
f[n_] := n!/Quotient[n, 2]!^2; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 85}] (* Jean-François Alcover, Jul 11 2017 *)
PROG
A246661_list = lambda len: RLT(lambda n: factorial(n)/factorial(n//2)^2, len)
(Python)
from math import factorial
def A246661(n): return RLT(n, lambda m: factorial(m)//factorial(m//2)**2) # Chai Wah Wu, Feb 04 2022
Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function ( A056040).
+20
7
1, 1, 1, 3, 23, 71, 757, 2559, 30671, 1383331, 5003791, 245273927, 3362110459, 12517624987, 175179377183, 9356953451851, 509614686432899, 1938763632210843, 107752663194272623
EXAMPLE
The 5th prime is 11, (11-1)$ = 252, the remainder term is (-1)^floor((11+2)/2)=1. So the quotient (252+1)/11 = 23 is the 5th member of the sequence.
MAPLE
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:
WQ := proc(f, r, n) map(p->(f(p-1)+r(p))/p, select(isprime, [$1..n])) end:
A163210 := n -> WQ(swing, p->(-1)^iquo(p+2, 2), n);
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 28 2013 *) a[p_] := (Binomial[p-1, (p-1)/2] - (-1)^((p-1)/2)) / p
Join[{1, 1}, a[Prime[Range[3, 20]]]] (* Peter Luschny, May 13 2017 *)
PROG
(PARI) a(n, p=prime(n)) = ((p-1)!/((p-1)\2)!^2 - (-1)^(p\2))/p \\ David A. Corneth, May 13 2017
Odd part of the swinging factorial A056040.
+20
7
1, 1, 1, 3, 3, 15, 5, 35, 35, 315, 63, 693, 231, 3003, 429, 6435, 6435, 109395, 12155, 230945, 46189, 969969, 88179, 2028117, 676039, 16900975, 1300075, 35102025, 5014575, 145422675, 9694845, 300540195, 300540195, 9917826435, 583401555, 20419054425, 2268783825
COMMENTS
Let n$ denote the swinging factorial. a(n) = n$ / 2^sigma(n) where sigma(n) is the exponent of 2 in the prime-factorization of n$. sigma(n) can be computed as the number of '1's in the base 2 representation of floor(n/2).
If n is even then a(n) is the numerator of the reduced ratio (n-1)!!/n!! = A001147(n-1)/ A000165(n), and if n is odd then a(n) is the numerator of the reduced ratio n!!/(n-1)!! = A001147(n)/ A000165(n-1). The denominators for each ratio should be compared to A060818. Here all ratios are reduced. - Anthony Hernandez, Feb 05 2020 [See the Mathematica program for a more compact form of the formula. Peter Luschny, Mar 01 2020 ]
FORMULA
a(n) = a(n-1)*n^((-1)^(n+1))*2^valuation(n, 2) for n > 0. - Peter Luschny, Sep 29 2019
EXAMPLE
11$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 3^2*7*11 = 2772/4 = 693.
a(7) = numerator((1*3*5*7)/(2*4*6)) = 35;
a(8) = numerator((1*3*5*7)/(2*4*6*8)) = 35;
a(9) = numerator((1*3*5*7*9)/(2*4*6*8)) = 315;
a(10) = numerator((1*3*5*7*9)/(2*4*6*8*10)) = 63. (End)
MAPLE
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:
sigma := n -> 2^(add(i, i= convert(iquo(n, 2), base, 2))):
a := n -> swing(n)/sigma(n);
MATHEMATICA
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/ f!]; a[n_] := With[{s = sf[n]}, s/2^IntegerExponent[s, 2]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *) r[n_] := (n - Mod[n - 1, 2])!! /(n - 1 + Mod[n - 1, 2])!! ;
Table[r[n], {n, 0, 36}] // Numerator (* Peter Luschny, Mar 01 2020 *)
PROG
@CachedFunction
def swing(n):
if n == 0: return 1
return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
# Alternatively:
(Sage)
@cached_function
if n == 0: return 1
return A163590(n - 1) * n^((-1)^(n + 1)) * 2^valuation(n, 2) (PARI)
my(a = vector(n+1)); a[1] = 1;
for(n = 1, n,
a[n+1] = a[n]*n^((-1)^(n+1))*2^valuation(n, 2));
Triangle interpolating the swinging factorial ( A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients ( A002426) with the central binomial coefficients ( A000984).
+20
6
1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870
COMMENTS
Triangle read by rows. For n >= 0, k >= 0 let T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i ( A056040). This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019
EXAMPLE
Triangle begins
1;
1, 2;
3, 4, 6;
7, 10, 14, 20;
19, 26, 36, 50, 70;
51, 70, 96, 132, 182, 252;
141, 192, 262, 358, 490, 672, 924;
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts: 1 1 3 7 19 51 ...
2 4 10 26 70 192 ...
6 14 36 96 262 726 ...
20 50 132 358 988 2760 ...
70 182 490 1346 3748 10540 ...
252 672 1836 5094 14288 40404 ...
(...)
Read by falling antidiagonals this yields the same sequence. (End)
MAPLE
For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle. a := n -> DiffTria(k->swing(2*k), n, true);
MATHEMATICA
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[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Swinging primes: primes which are within 1 of a swinging factorial ( A056040).
+20
5
2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011, 48619, 51479, 51481, 2704157, 155117519, 280816201, 4808643121, 35345263801, 81676217699, 1378465288199, 2104098963721, 5651707681619, 94684453367401, 386971244197199, 1580132580471899, 1580132580471901
EXAMPLE
3$ + 1 = 7 is prime, so 7 is in the sequence. (Here '$' denotes the swinging factorial function.)
MAPLE
# Seq with arguments <= n:
a := proc(n) select(isprime, map(x -> A056040(x)+1, [$1..n])); select(isprime, map(x -> A056040(x)-1, [$1..n])); sort(convert(convert(%%, set) union convert(%, set), list)) end:
MATHEMATICA
Reap[Do[f = n!/Quotient[n, 2]!^2; If[PrimeQ[p = f - 1], Sow[p]]; If[PrimeQ[p = f + 1], Sow[p]], {n, 1, 45}]][[2, 1]] // Union (* Jean-François Alcover, Jun 28 2013 *)
Primes of the form k$ + 1. Here '$' denotes the swinging factorial function ( A056040).
+20
5
2, 3, 7, 31, 71, 631, 3433, 51481, 2704157, 280816201, 4808643121, 35345263801, 2104098963721, 94684453367401, 1580132580471901, 483701705079089804581, 6892620648693261354601, 410795449442059149332177041, 2522283613639104833370312431401
EXAMPLE
Since 3$ = 4$ = 6 the prime 7 is listed, however only once.
MAPLE
a := proc(n) select(isprime, map(x -> A056040(x)+1, [$1..n])) end:
MATHEMATICA
Reap[Do[f = n!/Quotient[n, 2]!^2; If[PrimeQ[p = f + 1], Sow[p]], {n, 1, 70}]][[2, 1]] // Union (* Jean-François Alcover, Jun 28 2013 *)
Triangle interpolating the swinging factorial ( A056040) restricted to odd indices with its binomial inverse.
+20
5
1, 5, 6, 19, 24, 30, 67, 86, 110, 140, 227, 294, 380, 490, 630, 751, 978, 1272, 1652, 2142, 2772, 2445, 3196, 4174, 5446, 7098, 9240, 12012, 7869, 10314, 13510, 17684, 23130, 30228, 39468, 51480
COMMENTS
Triangle read by rows. For n >= 0, k >= 0 let
T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i ( A056040).
EXAMPLE
1
5, 6
19, 24, 30
67, 86, 110, 140
227, 294, 380, 490, 630
751, 978, 1272, 1652, 2142, 2772
2445, 3196, 4174, 5446, 7098, 9240, 12012
MAPLE
For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle. a := n -> DiffTria(k->swing(2*k+1), n, true);
MATHEMATICA
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[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
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