A001464 Expansion of e.g.f. exp(-x - (1/2)*x^2).
1, -1, 0, 2, -2, -6, 16, 20, -132, -28, 1216, -936, -12440, 23672, 138048, -469456, -1601264, 9112560, 18108928, -182135008, -161934624, 3804634784, -404007680, -83297957568, 92590134208, 1906560847424, -4221314202624, -45349267830400, 159324751301248
Offset: 0
Examples
G.f. = 1 - x + 2*x^3 - 2*x^4 - 6*x^5 + 16*x^6 + 20*x^7 - 132*x^8 + ...
References
- Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, New York, 1945, page 32.
- 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
- Robert Israel, Table of n, a(n) for n = 0..807
- John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.
- Robert Israel, Solution to Problem 91-9: Even Minus Odd Involutions in the Symmetric Group, SIAM Review 34(2)(1992), 315-317.
- L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
- Eric Weisstein's World of Mathematics, Bell Polynomial.
Crossrefs
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(-x-x^2/2) ))); // G. C. Greubel, Sep 03 2023 -
Maple
f:= gfun:-rectoproc({a(n)=-a(n-1)-(n-1)*a(n-2), a(0)=1,a(1)=-1},a(n),remember): map(f, [$0..100]); # Robert Israel, Apr 27 2017 a := n -> (-1)^n*2^((n-1)/2)*KummerU((1-n)/2, 3/2, 1/2): seq(simplify(a(n)), n=0..28); # Peter Luschny, Apr 30 2017 -
Mathematica
With[{nn=30},CoefficientList[Series[Exp[-x-1/2 x^2],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Sep 16 2011 *) a[ n_] := If[ n < 0, 0, HermiteH[ n, Sqrt[1/2]] (-Sqrt[1/2])^n]; (* Michael Somos, Jan 24 2014 *) a[ n_] := If[ n < 0, 0, (-1)^n Sum[ (-1)^k Binomial[ n, 2 k] (2 k - 1)!!, {k, 0, n/2}]]; (* Michael Somos, Jan 24 2014 *) Table[(-1)^(n + 1)*DifferenceRoot[Function[{y, m}, {y[1 + m] == y[m] - (n - m) y[m - 1], y[0] == 0, y[1] == 1, y[2] == 1}]][n], {n, 1, 30}] (* Benedict W. J. Irwin, Nov 03 2016 *) -
PARI
Vec( serlaplace( exp( -x -(1/2)*x^2 + O(x^66) ) ) ) /* Joerg Arndt, Oct 13 2012 */
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PARI
{a(n) = if( n<0, 0, (-1)^n * sum(k=0, n\2, (-1/2)^k * n! / (k! * (n - 2*k)!)))}; /* Michael Somos, Jan 24 2014 */ -
SageMath
def A001464_list(prec): P.
= PowerSeriesRing(QQ, prec) return P( exp(-x-x^2/2) ).egf_to_ogf().list() A001464_list(40) # G. C. Greubel, Sep 03 2023
Formula
From Benoit Cloitre, May 01 2003: (Start)
a(n) = -h(n, -1) where h(n, x) is the Hermite polynomial h(n, x) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n, 2*k)*Product_{i=0..k} (2*i-1)*x^(n-2*k).
a(n) = (-1)^n*Sum_{k=0..floor(n/2)} (-1)^k*C(n, 2*k)*(2k-1)!!. (End)
a(n) = -a(n-1) - (n-1)*a(n-2); a(0)=1, a(1)=-1. - Matthew J. White (mattjameswhite(AT)hotmail.com), Mar 01 2006
From Sergei N. Gladkovskii, Oct 12 2012, Nov 04 2012, Apr 17 2013, Nov 13 2013: (Start)
Continued fractions:
G.f.: 1/(U(0) + x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 + x/U(k+1)).
G.f.: 1/U(0) where U(k) = 1 + x + x^2*(k+1)/U(k+1).
G.f.: 1/Q(0) where Q(k) = 1 + x*k + x/(1 - x*(k+1)/Q(k+1)).
G.f.: T(0)/(1+x) where T(k) = 1 - x^2*(k+1)/(x^2*(k+1) + (1+x)^2/T(k+1)). (End)
From Michael Somos, Jan 24 2014: (Start)
Binomial transform is [1, 0, -1, 0, 3, 0, -15, 0, 105, ...] where A001147 = [1, 1, 3, 15, 105, ...].
Hankel transform is [1, -1, -2, 12, 288, -34560, -24883200, ...] where A000178 = [1, 1, 2, 12, 288, 34560, 24883200, ...].
0 = a(n) * (-a(n+1) - a(n+2) - a(n+3)) + a(n+1) * (a(n+1) + a(n+2)) for all n in Z. (End)
a(n) = -(-1)^n*y(n,n), where y(m+1,n) = y(m,n) - (n-m)*y(m-1,n), with y(0,n)=0, y(1,n)=y(2,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) = (-1)^n*2^((n-1)/2)*KummerU((1-n)/2, 3/2, 1/2). - Peter Luschny, Apr 30 2017
a(n) = Sum_{k=0..n} 2^k * Stirling1(n,k) * Bell_k(-1/2), where Bell_n(x) is n-th Bell polynomial. - Seiichi Manyama, Jan 31 2024
Extensions
a(12) and a(13) corrected by Simon Plouffe
Comments