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
Links
- 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.
Programs
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GAP
List([0..20], n-> 2^n*Factorial(n+2)); # G. C. Greubel, May 05 2019
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Magma
[2^n*Factorial(n+2): n in [0..20]]; // G. C. Greubel, May 05 2019
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Maple
seq(count(Permutation(n+1))*count(Composition(n)),n=1..17); # Zerinvary Lajos, Oct 16 2006
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Mathematica
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 *) -
PARI
a(n) = (n+2)!*2^n; \\ Joerg Arndt, May 05 2019
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Sage
[2^n*factorial(n+2) for n in (0..20)] # G. C. Greubel, May 05 2019
Formula
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)
Comments