|
%I M3957 N1631 #151 May 10 2022 14:20:15
%S 1,1,5,29,233,2329,27949,391285,6260561,112690097,2253801941,
%T 49583642701,1190007424825,30940193045449,866325405272573,
%U 25989762158177189,831672389061670049,28276861228096781665,1017967004211484139941,38682746160036397317757
%N Expansion of e.g.f. exp(-x)/(1-2*x).
%C a(n) is the permanent of the n X n matrix with 1's on the diagonal and 2's elsewhere. - Yuval Dekel, Nov 01 2003. Compare A157142.
%C Starting with offset 1 = lim_{k->infinity} M^k, where M = a tridiagonal matrix with (1,0,0,0,...) in the main diagonal, (1,3,5,7,...) in the subdiagonal and (2,4,6,8,...) in the subsubdiagonal. - _Gary W. Adamson_, Jan 13 2009
%C a(n) is also the number of (n-1)-dimensional facet derangements for the n-dimensional hypercube. - Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009
%C a(n) is the number of ways to write down each n-permutation and underline some (possibly none or all) of the elements that are not fixed points. a(n) = Sum_{k=0..n} A008290(n,k)*2^(n-k). - _Geoffrey Critzer_, Dec 15 2012
%C Type B derangement numbers: the number of fixed point free permutations in the n-th hyperoctahedral group of signed permutations of {1,2,...,n}. See Chow 2006. See A000166 for type A derangement numbers. - _Peter Bala_, Jan 30 2015
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000354/b000354.txt">Table of n, a(n) for n = 0..100</a>
%H Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry4/barry271.html">General Eulerian Polynomials as Moments Using Exponential Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.9.6.
%H Chak-On Chow, <a href="http://emis.ams.org/journals/SLC/wpapers/s55chow.html">On derangement polynomials of type B</a>, Séminaire Lotharingien de Combinatoire 55 (2006), Article B55b.
%H Gary Gordon and Elizabeth McMahon, <a href="http://arxiv.org/abs/0906.4253">Moving faces to other places: Facet derangements</a>, arXiv:0906.4253 [math.CO], 2009.
%H Gary Gordon and Elizabeth McMahon, <a href="http://www.jstor.org/stable/10.4169/000298910X523353">Moving faces to other places: facet derangements</a>, Amer. Math. Monthly, 117 (2010), 865-88.
%H Édouard Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k29021h">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1, p. 223.
%H Édouard Lucas, <a href="/A000899/a000899.pdf">Théorie des nombres</a> (annotated scans of a few selected pages)
%H István Mezo, Victor H. Moll, José L. Ramírez, and Diego Villamizar, <a href="https://arxiv.org/abs/2103.04151">On the r-Derangements of type B</a>, arXiv:2103.04151 [math.CO], 2021.
%H István Mező, Victor H. Moll, José Ramírez, and Diego Villamizar, <a href="https://hosted.math.rochester.edu/ojac/vol16/243.pdf">On the r-derangements of type B</a>, Online Journal of Analytic Combinatorics, Issue 16 (2021), #05.
%H Simon Plouffe, <a href="http://plouffe.fr/simon/exact.htm">Exact formulas for integer sequences</a>
%H L. W. Shapiro & N. J. A. Sloane, <a href="/A006318/a006318_1.pdf">Correspondence, 1976</a>
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F Inverse binomial transform of double factorials A000165. - _Paul Barry_, May 26 2003
%F a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n, k)*k!*2^k. - _Paul Barry_, May 26 2003
%F a(n) = Sum_{k=0..n} A008290(n, k)*2^(n-k). - _Philippe Deléham_, Dec 13 2003
%F a(n) = 2*n*a(n-1) + (-1)^n, n > 0, a(0)=1. - _Paul Barry_, Aug 26 2004
%F D-finite with recurrence a(n) = (2*n-1)*a(n-1) + (2*n-2)*a(n-2). - Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009
%F From _Groux Roland_, Jan 17 201: (Start)
%F a(n) = (1/(2*sqrt(exp(1))))*Integral_{x=-1..infinity} exp(-x/2)*x^n dx;
%F Sum_{k>=0} 1/(k!*2^(k+1)*(n+k+1)) = (-1)^n*(a(n)*sqrt(exp(1))-2^n*n!). (End)
%F a(n) = round(2^n*n!/exp(1/2)), x >= 0. - _Simon Plouffe_, Mar 1993
%F G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Sep 30 2013
%F From _Peter Bala_, Jan 30 2015: (Start)
%F a(n) = Integral_{x = 0..inf} (2*x - 1)^n*exp(-x) dx.
%F b(n) := 2^n*n! satisfies the recurrence b(n) = (2*n - 1)*b(n-1) + (2*n - 2)*b(n-2), the same recurrence as satisfied by a(n). This leads to the continued fraction representation a(n) = 2^n*n!*( 1/(1 + 1/(1 + 2/(3 + 4/(5 +...+ (2*n - 2)/(2*n - 1) ))))) for n >= 2, which in the limit gives the continued fraction representation sqrt(e) = 1 + 1/(1 + 2/(3 + 4/(5 + ... ))). (End)
%F For n > 0, a(n) = 1 + 4*Sum_{k=0..n-1} A263895(n). - _Vladimir Reshetnikov_, Oct 30 2015
%F a(n) = (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)). - _Peter Luschny_, May 09 2017
%F a(n+1) >= A113012(n). - _Michael Somos_, Sep 28 2017
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2*k - 1) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020
%F a(n) = 2^n*KummerU(-n, -n, -1/2). - _Peter Luschny_, May 10 2022
%e G.f. = 1 + x + 5*x^2 + 29*x^3 + 233*x^4 + 2329*x^5 + 27949*x^6 + 391285*x^7 + ... - _Michael Somos_, Apr 14 2018
%p a := n -> (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)):
%p seq(simplify(a(n)), n=0..18); # _Peter Luschny_, May 09 2017
%p a := n -> 2^n*add((n!/k!)*(-1/2)^k, k=0..n):
%p seq(a(n), n=0..23); # _Peter Luschny_, Jan 06 2020
%p seq(simplify(2^n*KummerU(-n, -n, -1/2)), n = 0..19); # _Peter Luschny_, May 10 2022
%t FunctionExpand @ Table[ Gamma[ n+1, -1/2 ]*2^n/Exp[ 1/2 ], {n, 0, 24}]
%t With[{nn=20},CoefficientList[Series[Exp[-x]/(1-2x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 22 2013 *)
%t a[n_] := 2^n n! Sum[(-1)^i/(2^i i!), {i, 0, n}]; Table[a[n], {n, 0, 20}] (* _Gerry Martens_ , May 06 2016 *)
%t a[ n_] := If[ n < 1, Boole[n == 0], (2 n - 1) a[n - 1] + (2 n - 2) a[n - 2]]; (* _Michael Somos_, Sep 28 2017 *)
%t a[ n_] := Sum[ (-1)^(n + k) Binomial[n, k] k! 2^k, {k, 0, n}]; (* _Michael Somos_, Apr 14 2018 *)
%t a[ n_] := If[ n < 0, 0, (2^n Gamma[n + 1, -1/2]) / Sqrt[E] // FunctionExpand]; (* _Michael Somos_, Apr 14 2018 *)
%o (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(-x)/(1-2*x))) \\ _Joerg Arndt_, Apr 15 2013
%o (PARI) vector(100, n, n--; sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k)) \\ _Altug Alkan_, Oct 30 2015
%o (PARI) {a(n) = if( n<1, n==0, (2*n - 1) * a(n-1) + (2*n - 2) * a(n-2))}; /* _Michael Somos_, Sep 28 2017 */
%Y Cf. A061714, A008290, A000166, A113012, A263895.
%Y Column k=2 of A320032.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
|
