A000354 Expansion of e.g.f. exp(-x)/(1-2*x).
1, 1, 5, 29, 233, 2329, 27949, 391285, 6260561, 112690097, 2253801941, 49583642701, 1190007424825, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 28276861228096781665, 1017967004211484139941, 38682746160036397317757
Offset: 0
Examples
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
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
- Paul 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)
- István Mezo, Victor H. Moll, José L. Ramírez, and Diego Villamizar, On the r-Derangements of type B, arXiv:2103.04151 [math.CO], 2021.
- István Mező, Victor H. Moll, José Ramírez, and Diego Villamizar, On the r-derangements of type B, Online Journal of Analytic Combinatorics, Issue 16 (2021), #05.
- Jean-Christophe Pain, A sum rule for r-derangements obtained from the Cauchy product of exponential generating functions, arXiv:2408.15927 [math.CO], 2024.
- 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.
Programs
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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 seq(simplify(2^n*KummerU(-n, -n, -1/2)), n = 0..19); # Peter Luschny, May 10 2022
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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 *) a[n_] := n! 2^n Hypergeometric1F1[-n, -n, -1/2]; Table[a[n], {n, 0, 19}] (* Peter Luschny, Jul 28 2024 *) -
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
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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 */
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 2011: (Start)
a(n) = (1/(2*sqrt(exp(1))))*Integral_{x>=-1} 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
a(n) = 2^n*KummerU(-n, -n, -1/2). - Peter Luschny, May 10 2022
a(n) = 2^n*n!*hypergeom([-n], [-n], -1/2). - Peter Luschny, Jul 28 2024
Comments