A077611 -id:A077611 - 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

A077613 Number of adjacent pairs of form (even,odd) among all permutations of {1,2,...,n}. Also, number of adjacent pairs of form (odd,even).

Original entry on oeis.org

0, 1, 4, 24, 144, 1080, 8640, 80640, 806400, 9072000, 108864000, 1437004800, 20118067200, 305124019200, 4881984307200, 83691159552000, 1506440871936000, 28810681675776000, 576213633515520000, 12164510040883200000, 267619220899430400000, 6182004002776842240000

Offset: 1

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

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..449

Cf. A000142, A002620, A077611, A077612.

Cf. A001620, A068985, A099284.

Array[Floor[#/2] Ceiling[#/2] (# - 1)! &, 19] (* Michael De Vlieger, Aug 16 2017 *)

a(n) = floor(n/2)*ceil(n/2)*(n-1)!; \\ Michel Marcus, Aug 29 2013

a(n) = floor(n/2)*ceiling(n/2)*(n-1)!. Proof: There are floor(n/2)*ceiling(n/2) pairs (r, s) with r even and s odd. 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) = A002620(n) * A000142(n-1). - Michel Marcus, Aug 29 2013
Sum_{n>=2} 1/a(n) = 6*(CoshIntegral(1) - gamma) + 2/e - 1 = 6*(A099284 - A001620) + 2*A068985 - 1. - Amiram Eldar, Jan 22 2023

A077612 Number of adjacent pairs of form (even,even) among all permutations of {1,2,...,n}.

Original entry on oeis.org

0, 0, 0, 12, 48, 720, 4320, 60480, 483840, 7257600, 72576000, 1197504000, 14370048000, 261534873600, 3661488230400, 73229764608000, 1171676233728000, 25609494822912000, 460970906812416000, 10948059036794880000, 218961180735897600000, 5620003638888038400000

Offset: 1

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

nonn

Cf. A000142, A077611, A077613, A110660.

Cf. A001113, A001620, A099284.

a[n_] := Floor[n/2]*Floor[n/2 - 1]*(n - 1)!; Array[a, 25] (* Amiram Eldar, Jan 22 2023 *)

a(n) = n\2 * (n\2-1)*(n-1)! ; \\ Michel Marcus, Aug 29 2013

a(n) = floor(n/2)*floor(n/2-1)*(n-1)!. Proof: There are floor(n/2)*floor(n/2-1) pairs (r, s) with r and s even 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) = A110660(n+2) * A000142(n-1). - Michel Marcus, Aug 29 2013
Sum_{n>=4} 1/a(n) = CoshIntegral(1) - gamma - 3*e + 8 = A099284 - A001620 - 3*A001113 + 8. - Amiram Eldar, Jan 22 2023

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)

cp's OEIS Frontend

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

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A077613 Number of adjacent pairs of form (even,odd) among all permutations of {1,2,...,n}. Also, number of adjacent pairs of form (odd,even).

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Mathematica

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A077612 Number of adjacent pairs of form (even,even) among all permutations of {1,2,...,n}.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

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Maple

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