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Search: a083746 -id:a083746
Displaying 1-4 of 4 results found.
page 1
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A052560
a(n) = 3*n!.
+10
12
3, 3, 6, 18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400, 1437004800, 18681062400, 261534873600, 3923023104000, 62768369664000, 1067062284288000, 19207121117184000, 364935301226496000, 7298706024529920000, 153272826515128320000
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
a(n) is the size of the centralizer of a 3-cycle in the symmetric group S_(n+3). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
3 times factorial numbers. - Omar E. Pol, Jan 17 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 502
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
Index entries for sequences related to factorial base representation
FORMULA
E.g.f.: 3/(1-x).
a(n) = n*a(n-1), with a(0) = 3.
For n>0: a(n) = Sum_{k=1..n+1} A083746(k). - Reinhard Zumkeller, Apr 14 2007
MAPLE
spec := [S, {S=Union(Sequence(Z), Sequence(Z), Sequence(Z))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
a:=n->sum(n!, k=1..n):seq(a(n)-sum(n!, k=4..n), n=0..19); # Zerinvary Lajos, Dec 22 2008
MATHEMATICA
3*Range[0, 25]! (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *)
PROG
(Magma) [3*Factorial(n): n in [0..25]]; // Vincenzo Librandi, Jun 13 2011
(PARI) a(n)=3*n! \\ Charles R Greathouse IV, Nov 20 2011
(Sage) [3*factorial(n) for n in range(25)] # G. C. Greubel, May 05 2019
CROSSREFS
Cf. A000142, A129380.
Row 13 of A276955 (from term a(3)=18 onward).
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved
A052673
a(n) = 3*n*n!.
+10
4
0, 3, 12, 54, 288, 1800, 12960, 105840, 967680, 9797760, 108864000, 1317254400, 17244057600, 242853811200, 3661488230400, 58845346560000, 1004293914624000, 18140058832896000, 345728180109312000
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 621
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
FORMULA
E.g.f.: 3*x/(1-x)^2.
Recurrence: a(0)=0, a(1)=3, (n-1)*a(n) = n^2*a(n-1).
a(n) = A122972(n+2) - A122972(n) for n > 0. - Reinhard Zumkeller, Sep 21 2006
For n>0: a(n) = A083746(n+2). - Reinhard Zumkeller, Apr 14 2007
G.f.: 3*Hypergeometric2F0([2,2], [], x). - G. C. Greubel, Jun 12 2022
MAPLE
spec := [S, {S=Prod(Sequence(Z), Sequence(Z), Union(Z, Z, Z))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
Table[3 n n!, {n, 0, 20}] (* Harvey P. Dale, Feb 12 2017 *)
PROG
(Magma) [3*(Factorial(n+1)-Factorial(n)): n in [0..30]]; // G. C. Greubel, Jun 12 2022
(SageMath) [3*n*factorial(n) for n in (0..30)] # G. C. Greubel, Jun 12 2022
CROSSREFS
Cf. A083746, A122972.
Cf. A000142, A008585.
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved
A082425
a(1)=1, a(n) = -1 + n*Sum_{j=1..n-1} a(j).
+10
3
1, 1, 5, 27, 169, 1217, 9939, 90871, 920069, 10222989, 123698167, 1619321459, 22805443881, 343835923129, 5525934478859, 94309281772527, 1703461402016269, 32465970250192421, 651123070017747999, 13707854105636799979
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..445
FORMULA
For n >= 2, a(n) = floor(n*(3-e)*n!).
a(n) = n*A056543(n) - 1, n > 1. - Vladeta Jovovic, Apr 26 2003
From Peter Bala, Jul 09 2008: (Start)
In the following remarks we use an offset of 1, i.e., a(1) = 1, a(2) = 1, a(3) = 5, ... .
For n >= 2, a(n) = n*n!*Sum_{k = 2..n} 1/(k*(k-1)*k!).
For n >= 2, a(n) = 3*n*n! - Sum_{k = 0..n} (k+1)!*binomial(n,k).
Limit_{n -> oo} a(n)/(n*n!) = 3 - e.
E.g.f.: 1 + t + (3*t - exp(t))/(1-t)^2.
a(n) = A083746(n+2) - A001339(n).
Recurrence relation: a(1) = 1, a(2) = 1, a(3) = 5, a(n) = (n+2)*a(n-1) - (n-1)*a(n-2) for n >= 4.
Recurrence relation: a(1) = 1, a(2) = 1, a(n) = (n^2*a(n-1) + 1)/(n-1) for n >= 2.
The recurrence relation x(n) = (n^2*x(n-1) - 1)/(n-1), for n >= 2, has the general solution x(n) = n*n!*x(1) - a(n); particular solutions are A007808 (x(1) = 1) and A001339 (x(1) = 3). (End)
MAPLE
a:= n -> n*n!*add(1/(k*(k-1)*k!), k = 2..n): seq(a(n), n = 2..20); # Peter Bala, Jul 09 2008
MATHEMATICA
a[n_]:= a[n]= If[n<3, 1, -1 +n*Sum[a[j], {j, n-1}]];
Table[a[n], {n, 40}] (* G. C. Greubel, Feb 03 2024 *)
PROG
(Magma) [n le 2 select 1 else (n^2*Self(n-1) +1)/(n-1): n in [1..30]]; // G. C. Greubel, Feb 03 2024
(SageMath)
@CachedFunction # a = A082425
def a(n): return 1 if (n==1) else -1 + n*sum(a(j) for j in range(1, n))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 03 2024
CROSSREFS
Cf. A001339, A007808, A056543, A074143, A083746.
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Apr 24 2003
EXTENSIONS
Offset corrected by G. C. Greubel, Feb 03 2024
STATUS
approved
A129379
a(n) = sum of sums of all sets of three distinct preceding terms otherwise a(n) = n for n <= 3.
+10
3
1, 2, 3, 6, 36, 288, 3360, 55440, 1241856, 36427776, 1358235648, 62818398720, 3531789972480, 237336286150656, 18792718657929216, 1732062236305809408, 183865068161693614080, 22273939685873740677120
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..270
FORMULA
a(n) = (1/2)*(n-2)*(n-3)*Sum_{j=1..n-1} a(j) for n > 3, with a(1) = 1, a(2) = 2, a(3) = 3.
a(n) = A000217(n-3)*A129380(n-1) for n > 3.
From G. C. Greubel, Feb 02 2024: (Start)
a(n) = (6/2^(n-4))*binomial(n-2,2)*|Pochhammer((3+i*sqrt(7))/2, n-4)|^2 for n > 3, otherwise a(n) = n.
a(n) = (3/2^(n-3))*binomial(n-2,2)*Product_{k=0..n-3} (k^2 - k + 2), for n > 3, otherwise a(n) = n.
a(n) = (n-2)*(n^2-7*n+14)/(2*(n-4))*a(n-1), for n > 4, otherwise a(n) = binomial(n, floor(n/2)).
(End)
From Vaclav Kotesovec, Feb 03 2024: (Start)
For n>=4, a(n) = 3 * 2^(3-n) * (n-3) * (n-2) * cosh(sqrt(7)*Pi/2) * Gamma(n - 5/2 - i*sqrt(7)/2) * Gamma(n - 5/2 + i*sqrt(7)/2)/Pi, where i is the imaginary unit.
a(n) ~ 3 * cosh(sqrt(7)*Pi/2) * n^(2*n-4) / (2^(n-4) * exp(2*n)). (End)
MATHEMATICA
a[n_]:= a[n]= If[n<5, Binomial[n, Floor[n/2]], (n-2)*(n^2-7*n+14)*a[n- 1]/(2*(n-4))];
Table[a[n], {n, 40}] (* G. C. Greubel, Feb 02 2024 *)
Round[Flatten[{{1, 2, 3}, Table[3 * 2^(3-n) * (n-3) * (n-2) * Cosh[Sqrt[7]*Pi/2] * Gamma[n - 5/2 - I*Sqrt[7]/2] * Gamma[n - 5/2 + I*Sqrt[7]/2]/Pi, {n, 4, 20}]}]] (* Vaclav Kotesovec, Feb 03 2024 *)
PROG
(Magma)
A129379:= func< n | n le 3 select Binomial(n, Floor(n/2)) else (3/2^(n-3))*Binomial(n-2, 2)*(&*[k^2-k+2: k in [0..n-3]]) >;
[A129379(n): n in [1..30]]; // G. C. Greubel, Feb 02 2024
(SageMath)
def A129379(n): return binomial(n, n//2) if n<4 else 3*binomial(n-2, 2)*product(j^2-j+2 for j in range(n-2))//2^(n-3)
[A129379(n) for n in range(1, 31)] # G. C. Greubel, Feb 02 2024
CROSSREFS
Cf. A000217, A083746, A129380.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 14 2007
STATUS
approved
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