A052618 -id:A052618 - cp's OEIS frontend

A052558 a(n) = n! ((-1)^n + 2n + 3)/4.

1, 1, 4, 12, 72, 360, 2880, 20160, 201600, 1814400, 21772800, 239500800, 3353011200, 43589145600, 697426329600, 10461394944000, 188305108992000, 3201186852864000, 64023737057280000, 1216451004088320000, 26761922089943040000, 562000363888803840000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Stirling transform of (-1)^(n+1)*a(n-1) = [1, -1, 4, -12, 72, -360, ...] is A052841(n-1) = [1,0,2,6,38,270,...]. - Michael Somos, Mar 04 2004
The Stirling transform of this sequence is A258369. - Philippe Deléham, May 17 2005; corrected by Ilya Gutkovskiy, Jul 25 2018
Ignoring reflections, this is the number of ways of connecting n+2 equally-spaced points on a circle with a path of n+1 line segments. See A030077 for the number of distinct lengths. - T. D. Noe, Jan 05 2007
From Gary W. Adamson, Apr 20 2009: (Start)
Signed: (+ - - + + - - + +, ...) = eigensequence of triangle A002260.
Example: -360 = (1, 1, -1, -4, 12, 71) dot (1, -2, 3, -4, 5, -6) = (1, -2, -3, 16, 60, -432). (End)
a(n) is the number of odd fixed points in all permutations of {1, 2, ..., n+1}, Example: a(2)=4 because we have 1'23', 1'32, 312, 213', 231, and 321, where the odd fixed points are marked. - Emeric Deutsch, Jul 18 2009
a(n) is also the number of permutations of [n+1] starting with an even number. - Olivier Gérard, Nov 07 2011

T. D. Noe, Table of n, a(n) for n=0..100
A. Moreno Canadas, P. F. Fernandez Espinoza, and I. D. M. Gaviria, Categorification of some integer sequences via Kronecker Modules, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 500.

Cf. A174548, A052841.
Cf. A002260. - Gary W. Adamson, Apr 20 2009
Cf. A052591. - Emeric Deutsch, Jul 18 2009
Cf. A052618, A077611, A199495. - Olivier Gérard, Nov 07 2011

List([0..30], n-> ((-1)^n +2*n +3)*Factorial(n)/4); # G. C. Greubel, May 07 2019

[((-1)^n +2*n +3)*Factorial(n)/4: n in [0..30]]; // G. C. Greubel, May 07 2019

spec := [S,{S=Prod(Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[n!((-1)^n+2n+3)/4,{n,0,30}] (* Harvey P. Dale, Aug 16 2014 *)

PARI
```
a(n)=if(n<0,0,(1+n\2)*n!)
```

a(n)=if(n<0, 0, n!*polcoeff(1/(1-x)/(1-x^2)+x*O(x^n), n))

[((-1)^n +2*n +3)*factorial(n)/4 for n in (0..30)] # G. C. Greubel, May 07 2019

D-finite with recurrence a(n) = a(n-1) + (n^2-1)*a(n-2), with a(1)=1, a(0)=1.
a(n) = ((-1)^n + 2*n + 3)*n!/4.
Let u(1)=1, u(n) = Sum_{k=1..n-1} u(k)*k*(-1)^(k-1) then a(n) = abs(u(n+2)). - Benoit Cloitre, Nov 14 2003
E.g.f.: 1/((1-x)*(1-x^2)).
From Emeric Deutsch, Jul 18 2009: (Start)
a(n) = (n+1)!/2 if n is odd; a(n) = n!(n+2)/2 if n is even.
a(n) = (n+1)! - A052591(n). (End)
E.g.f.: G(0)/(1+x) where G(k) = 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)*(2*k+3)/(x*(2*k+3) + 2*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2012
Sum_{n>=0} 1/a(n) = e - 1/e = 2*sinh(1) (A174548). - Amiram Eldar, Jan 22 2023

A077611 Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,...,n}.

Original entry on oeis.org

0, 0, 4, 12, 144, 720, 8640, 60480, 806400, 7257600, 108864000, 1197504000, 20118067200, 261534873600, 4881984307200, 73229764608000, 1506440871936000, 25609494822912000, 576213633515520000, 10948059036794880000, 267619220899430400000, 5620003638888038400000

Offset: 1

Leroy Quet, Frank Ruskey, and Dean Hickerson, Nov 11 2002

nonn

a(n) is also the number of permutations of [n+1] starting and ending with an even number. - Olivier Gérard, Nov 07 2011

			For n=4, the a(4) = 12 permutations of degree 5 starting and ending with an even number are 21354, 21534, 23154, 23514, 25134, 25314, 41352, 41532, 43152, 43512, 45132, 45312.

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Cf. A052618, A077612, A077613.

Cf. A001620, A073742, A099284.

[Factorial(n-1)*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8 : n in [1..30]]; // Vincenzo Librandi, Nov 16 2011

Table[Ceiling[n/2] Ceiling[n/2 - 1] (n - 1)!, {n, 22}] (* Michael De Vlieger, Aug 20 2017 *)

a(n) = ceiling(n/2)*ceiling(n/2-1)*(n-1)!. Proof: There are ceiling(n/2) * ceiling(n/2-1) pairs (r, s) with r and s odd and distinct. For each pair, there are n-1 places it can occur in a permutation and (n-2)! possible arrangements of the other numbers.
a(n) = (n-1)!*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8. - Bruno Berselli, Nov 07 2011
Sum_{n>=3} 1/a(n) = 4*(CoshIntegral(1) - gamma - sinh(1) + 1) = 4*(A099284 - A001620 - A073742 + 1). - Amiram Eldar, Jan 22 2023

A152666 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of odd entries (1<=k<=ceiling(n/2)). For example, the permutation 321756498 has 3 runs of odd entries: 3, 175 and 9.

Original entry on oeis.org

1, 2, 4, 2, 12, 12, 36, 72, 12, 144, 432, 144, 576, 2592, 1728, 144, 2880, 17280, 17280, 2880, 14400, 115200, 172800, 57600, 2880, 86400, 864000, 1728000, 864000, 86400, 518400, 6480000, 17280000, 12960000, 2592000, 86400, 3628800, 54432000

Offset: 1

Emeric Deutsch, Dec 14 2008

Sum of entries in row n is n! (=A000142(n)).
Row n contains ceiling(n/2) entries.
T(n,1) = A010551(n+1).
Sum_{k>=1} k*T(n,k) = A052618(n-1).
Mirror image of A134435.

			T(3,2)=2 because we have 123 and 321.
T(4,2)=12 because we have 1234, 1432, 3214, 3412, 1243, 3241 and their reverses.
Triangle starts:
1;
2;
4,2;
12,12;
36,72,12;
144,432,144;
576,2592,1728,144.

Cf. A000142, A010551, A052618, A152667, A134435.

ae := proc (n, k) options operator, arrow: factorial(n)^2*binomial(n+1, k)*binomial(n-1, k-1) end proc: ao := proc (n, k) options operator, arrow: factorial(n)*factorial(n+1)*binomial(n, k-1)*binomial(n+1, k) end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc: for n to 12 do seq(T(n, k), k = 1 .. ceil((1/2)*n)) end do; # yields sequence in triangular form

T[n_?EvenQ, k_] := (n/2)!^2*Binomial[n/2 - 1, k - 1]*Binomial[n/2 + 1, k]; T[n_?OddQ, k_] := ((n - 1)/2 + 1)!*((n - 1)/2)!*Binomial[(n - 1)/2 + 1, k]*Binomial[(n - 1)/2, k - 1]; Table[T[n, k], {n, 1, 12}, {k, 1, Floor[(n + 1)/2]}] // Flatten (* Jean-François Alcover, Nov 13 2016 *)

T(2n,k) = (n!)^2*binomial(n+1,k)*binomial(n-1,k-1).

T(2n+1,k) = n!*(n+1)!*binomial(n,k-1)*binomial(n+1,k).

A199495 Number of permutations of [n] starting and ending with an odd number.

Original entry on oeis.org

0, 1, 0, 2, 4, 36, 144, 1440, 8640, 100800, 806400, 10886400, 108864000, 1676505600, 20118067200, 348713164800, 4881984307200, 94152554496000, 1506440871936000, 32011868528640000, 576213633515520000, 13380961044971520000, 267619220899430400000

Offset: 0

Olivier Gérard, Nov 07 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A052618, A077611, A152887.

a:= n-> `if`(n<2, n, ceil(n/2)*(ceil(n/2)-1)*(n-2)!):
seq(a(n), n=0..30); # Alois P. Heinz, Nov 07 2011

From Alois P. Heinz, Nov 07 2011: (Start)
a(n) = 2*(n-2)! * [x^(n-2)] x/((1+x)^2*(1-x)^3) for n>=2, else a(n) = n.
a(n) = 2*A152887(n-2) for n>=2, else a(n) = n. (End)

A152668 Number of runs of even entries in all permutations of {1,2,...,n} (the permutation 274831659 has 3 runs of even entries: 2, 48 and 6).

Original entry on oeis.org

2, 6, 36, 192, 1440, 10800, 100800, 967680, 10886400, 127008000, 1676505600, 22992076800, 348713164800, 5492232345600, 94152554496000, 1673823191040000, 32011868528640000, 633834996867072000, 13380961044971520000

Offset: 2

Emeric Deutsch, Dec 14 2008

nonn

a(n) = Sum(k*A152667(n,k), k=1..floor(n/2)).

			a(3) = 6 because each of the permutations 123, 132, 213, 231, 312, 321 has exactly 1 run of even entries.

Cf. A152667, A052618.

ae := proc (n) options operator, arrow: (1/2)*factorial(2*n)*(n+1) end proc: ao := proc (n) options operator, arrow: n*(n+2)*factorial(2*n) end proc: a := proc (n) if `mod`(n, 2) = 0 then ae((1/2)*n) else ao((1/2)*n-1/2) end if end proc; seq(a(n), n = 2 .. 20);

a[n_] := If[EvenQ[n], (n/2+1)n!/2, ((n-1)/2)((n-1)/2+2)(n-1)!];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Apr 09 2024 *)

a(2n) = (n+1)(2n)!/2;
a(2n+1) = n(n+2)(2n)!.
D-finite with recurrence a(n) -2*a(n-1) -n*(n-1)*a(n-2) +2*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

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A052558 a(n) = n! *((-1)^n + 2*n + 3)/4.

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A077611 Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,...,n}.

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A152666 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of odd entries (1<=k<=ceiling(n/2)). For example, the permutation 321756498 has 3 runs of odd entries: 3, 175 and 9.

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A199495 Number of permutations of [n] starting and ending with an odd number.

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A152668 Number of runs of even entries in all permutations of {1,2,...,n} (the permutation 274831659 has 3 runs of even entries: 2, 48 and 6).

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A052558 a(n) = n! ((-1)^n + 2n + 3)/4.