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 A000354 Expansion of e.g.f. exp(-x)/(1-2*x). (Formerly M3957 N1631) 23
 1, 1, 5, 29, 233, 2329, 27949, 391285, 6260561, 112690097, 2253801941, 49583642701, 1190007424825, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 28276861228096781665, 1017967004211484139941, 38682746160036397317757 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. Starting with offset 1 = lim_{k->inf.} 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 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 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 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 REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..100 Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013 P. Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6. Chak-On Chow,On derangement polynomials of type B, Séminaire Lotharingien de Combinatoire 55 (2006), Article B55b. Gary Gordon and Elizabeth McMahon, Moving faces to other places: Facet derangements, arXiv:0906.4253 [math.CO], 2009. Gary Gordon and Elizabeth McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88. Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 223. Édouard Lucas, Théorie des nombres (annotated scans of a few selected pages) Simon Plouffe, Exact formulas for integer sequences L. W. Shapiro & N. J. A. Sloane, Correspondence, 1976 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1. FORMULA Inverse binomial transform of double factorials A000165. - Paul Barry, May 26 2003 a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n, k)*k!*2^k. - Paul Barry, May 26 2003 a(n) = Sum_{k=0..n} A008290(n, k)*2^(n-k). - Philippe Deléham, Dec 13 2003 a(n) = 2*n*a(n-1) + (-1)^n, n > 0, a(0)=1. - Paul Barry, Aug 26 2004 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 From Groux Roland, Jan 17 201: (Start) a(n) = (1/(2*sqrt(exp(1))))*Integral_{x=-1..infinity} exp(-x/2)*x^n dx; Sum_{k>=0} 1/(k!*2^(k+1)*(n+k+1)) = (-1)^n*(a(n)*sqrt(exp(1))-2^n*n!). (End) a(n) = round(2^n*n!/exp(1/2)), x >= 0. - Simon Plouffe, Mar 1993 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 From Peter Bala, Jan 30 2015: (Start) a(n) = Integral_{x = 0..inf} (2*x - 1)^n*exp(-x) dx. 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) For n > 0, a(n) = 1 + 4*Sum_{k=0..n-1} A263895(n). - Vladimir Reshetnikov, Oct 30 2015 a(n) = (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)). - Peter Luschny, May 09 2017 a(n+1) >= A113012(n). - Michael Somos, Sep 28 2017 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2*k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020 EXAMPLE 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 MAPLE a := n -> (-1)^n*(1-2*n*hypergeom([1, 1-n], [], 2)): seq(simplify(a(n)), n=0..18); # Peter Luschny, May 09 2017 a := n -> 2^n*add((n!/k!)*(-1/2)^k, k=0..n): seq(a(n), n=0..23); # Peter Luschny, Jan 06 2020 MATHEMATICA FunctionExpand @ Table[ Gamma[ n+1, -1/2 ]*2^n/Exp[ 1/2 ], {n, 0, 24}] With[{nn=20}, CoefficientList[Series[Exp[-x]/(1-2x), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jul 22 2013 *) 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 *) 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 *) a[ n_] := Sum[ (-1)^(n + k) Binomial[n, k] k! 2^k, {k, 0, n}]; (* Michael Somos, Apr 14 2018 *) a[ n_] := If[ n < 0, 0, (2^n Gamma[n + 1, -1/2]) / Sqrt[E] // FunctionExpand]; (* Michael Somos, Apr 14 2018 *) PROG (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(-x)/(1-2*x))) \\ Joerg Arndt, Apr 15 2013 (PARI) vector(100, n, n--; sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k)) \\ Altug Alkan, Oct 30 2015 (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 */ CROSSREFS Cf. A061714, A008290, A000166, A113012, A263895. Column k=2 of A320032. Sequence in context: A309260 A087662 A113012 * A103815 A134752 A231712 Adjacent sequences:  A000351 A000352 A000353 * A000355 A000356 A000357 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified May 30 08:04 EDT 2020. Contains 334712 sequences. (Running on oeis4.)