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A000354
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Expansion of e.g.f. exp(-x)/(1-2*x).
(Formerly M3957 N1631)
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32
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1, 1, 5, 29, 233, 2329, 27949, 391285, 6260561, 112690097, 2253801941, 49583642701, 1190007424825, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 28276861228096781665, 1017967004211484139941, 38682746160036397317757
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OFFSET
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0,3
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COMMENTS
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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->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
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
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n, k)*k!*2^k. - Paul Barry, May 26 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
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)
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
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)
a(n) = (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)). - Peter Luschny, May 09 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*n!*hypergeom([-n], [-n], -1/2). - Peter Luschny, Jul 28 2024
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EXAMPLE
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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
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MAPLE
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a := n -> (-1)^n*(1-2*n*hypergeom([1, 1-n], [], 2)):
a := n -> 2^n*add((n!/k!)*(-1/2)^k, k=0..n):
seq(simplify(2^n*KummerU(-n, -n, -1/2)), n = 0..19); # Peter Luschny, May 10 2022
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MATHEMATICA
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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];
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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STATUS
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approved
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