A051578 -id:A051578 - cp's OEIS frontend

A014297 a(n) = n! * C(n+2, 2) * 2^(n+1).

2, 12, 96, 960, 11520, 161280, 2580480, 46448640, 928972800, 20437401600, 490497638400, 12752938598400, 357082280755200, 10712468422656000, 342798989524992000, 11655165643849728000, 419585963178590208000, 15944266600786427904000, 637770664031457116160000

Offset: 0

Emeric Deutsch

Partition the set {1,2,...,n+2} into an even number of subsets. Arrange (linearly order) the elements within each subset and then arrange the subsets. - Geoffrey Critzer, Mar 03 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 506
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

Essentially the same as A052564.

Cf. A088312.

List([0..20], n-> 2^n*Factorial(n+2)); # G. C. Greubel, May 05 2019

[2^n*Factorial(n+2): n in [0..20]]; // G. C. Greubel, May 05 2019

seq(count(Permutation(n+1))*count(Composition(n)),n=1..17); # Zerinvary Lajos, Oct 16 2006

Drop[CoefficientList[Series[(1-x)^2/(1-2x), {x, 0, 20}], x]* Table[n!, {n, 0, 20}], 2] (* Geoffrey Critzer, Mar 03 2010 *)
Part[#, Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0]*#)&@CoefficientList[Series[2/(1 - 2*x)^3, {x, 0, 20}], x]// ExpandAll (* Vincenzo Librandi, Jan 04 2013 - after Olivier Gérard in A213068 *)
Table[n!Binomial[n+2,2]2^(n+1),{n,0,30}] (* Harvey P. Dale, Dec 27 2022 *)

a(n) = (n+2)!*2^n; \\ Joerg Arndt, May 05 2019

[2^n*factorial(n+2) for n in (0..20)] # G. C. Greubel, May 05 2019

a(n) = Sum_{k=0..n} (n+2)!*C(n,k).
Prepend the sequence with 1,0, then e.g.f. is (1-x)^2/(1-2*x). - Geoffrey Critzer, Mar 03 2010
E.g.f.: 2/(1-2*x)^3. - R. J. Mathar, Nov 27 2011
a(n) = 2*A051578(n). - R. J. Mathar, Apr 26 2017
a(n) = (n+2)! * 2^n. - Joerg Arndt, May 05 2019
From Amiram Eldar, Jul 04 2020: (Start)
Sum_{n>=0} 1/a(n) = 4*sqrt(e) - 6.
Sum_{n>=0} (-1)^n/a(n) = 4/sqrt(e) - 2. (End)

A052587 Expansion of e.g.f. x^2(1-x)/(1-2x).

Original entry on oeis.org

0, 0, 2, 6, 48, 480, 5760, 80640, 1290240, 23224320, 464486400, 10218700800, 245248819200, 6376469299200, 178541140377600, 5356234211328000, 171399494762496000, 5827582821924864000, 209792981589295104000

Offset: 0

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

Essentially the same as A051578. - R. J. Mathar, Feb 22 2008

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 532

Cf. A051578 (essentially the same), A000165 (double factorial), A000142 (factorial), A000079 (powers of 2).

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

With[{nn=20},CoefficientList[Series[x^2 (1-x)/(1-2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 14 2025 *)

apply( {A052587(n)=n!<<(n-3)+(n==2)}, [0..19]) \\ M. F. Hasler, Nov 10 2024

E.g.f.: x^2*(-1+x)/(-1+2*x).
Recurrence: a(0) = a(1) = 0, a(2) = 2, a(3) = 6, a(n+1) = 2*(n+1)*a(n) for n > 3.
a(n) = 1/8*2^n*n!, n > 2.
a(n) = A051578(n-2) = (2*n)!!/4!! for n > 2. - M. F. Hasler, Nov 10 2024

cp's OEIS Frontend

A014297 a(n) = n! * C(n+2, 2) * 2^(n+1).

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Magma

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A052587 Expansion of e.g.f. x^2(1-x)/(1-2x).

Views

Author

Keywords

Comments

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

cp's OEIS Frontend

A014297 a(n) = n! * C(n+2, 2) * 2^(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A052587 Expansion of e.g.f. x^2*(1-x)/(1-2*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A052587 Expansion of e.g.f. x^2(1-x)/(1-2x).