A001464 - OEIS

%I M0361 N0137 #100 Jan 31 2024 10:08:43

%S 1,-1,0,2,-2,-6,16,20,-132,-28,1216,-936,-12440,23672,138048,-469456,

%T -1601264,9112560,18108928,-182135008,-161934624,3804634784,

%U -404007680,-83297957568,92590134208,1906560847424,-4221314202624,-45349267830400,159324751301248

%N Expansion of e.g.f. exp(-x - (1/2)*x^2).

%C From _Robert Israel_, Apr 27 2017: (Start)

%C (-1)^n*a(n) is (the number of even involutions) - (the number of odd involutions) in the symmetric group S_n.

%C a(n) == (-1)^n (mod A069834(n-1)) for n >= 3.

%C a(n) is divisible by n-2 and by A200675(n+2). (End)

%D Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, New York, 1945, page 32.

%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 Robert Israel, <a href="/A001464/b001464.txt">Table of n, a(n) for n = 0..807</a>

%H John Campbell, <a href="https://dmtcs.episciences.org/3210">A class of symmetric difference-closed sets related to commuting involutions</a>, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.

%H Robert Israel, <a href="http://dx.doi.org/10.1137/1034062">Solution to Problem 91-9: Even Minus Odd Involutions in the Symmetric Group</a>, SIAM Review 34(2)(1992), 315-317.

%H L. Moser and M. Wyman, <a href="http://cms.math.ca/10.4153/CJM-1955-021-8">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F From _Benoit Cloitre_, May 01 2003: (Start)

%F 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).

%F a(n) = (-1)^n*Sum_{k=0..floor(n/2)} (-1)^k*C(n, 2*k)*(2k-1)!!. (End)

%F 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

%F From _Sergei N. Gladkovskii_, Oct 12 2012, Nov 04 2012, Apr 17 2013, Nov 13 2013: (Start)

%F Continued fractions:

%F G.f.: 1/(U(0) + x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 + x/U(k+1)).

%F G.f.: 1/U(0) where U(k) = 1 + x + x^2*(k+1)/U(k+1).

%F G.f.: 1/Q(0) where Q(k) = 1 + x*k + x/(1 - x*(k+1)/Q(k+1)).

%F 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)

%F From _Michael Somos_, Jan 24 2014: (Start)

%F Binomial transform is [1, 0, -1, 0, 3, 0, -15, 0, 105, ...] where A001147 = [1, 1, 3, 15, 105, ...].

%F Hankel transform is [1, -1, -2, 12, 288, -34560, -24883200, ...] where A000178 = [1, 1, 2, 12, 288, 34560, 24883200, ...].

%F 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)

%F 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

%F a(n) = (-1)^n*2^((n-1)/2)*KummerU((1-n)/2, 3/2, 1/2). - _Peter Luschny_, Apr 30 2017

%F 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

%e G.f. = 1 - x + 2*x^3 - 2*x^4 - 6*x^5 + 16*x^6 + 20*x^7 - 132*x^8 + ...

%p f:= gfun:-rectoproc({a(n)=-a(n-1)-(n-1)*a(n-2), a(0)=1,a(1)=-1},a(n),remember):

%p map(f, [$0..100]); # _Robert Israel_, Apr 27 2017

%p 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

%t With[{nn=30},CoefficientList[Series[Exp[-x-1/2 x^2],{x,0,nn}], x]Range[0,nn]!] (* _Harvey P. Dale_, Sep 16 2011 *)

%t a[ n_] := If[ n < 0, 0, HermiteH[ n, Sqrt[1/2]] (-Sqrt[1/2])^n]; (* _Michael Somos_, Jan 24 2014 *)

%t 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 *)

%t 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 *)

%o (PARI) Vec( serlaplace( exp( -x -(1/2)*x^2 + O(x^66) ) ) ) /* _Joerg Arndt_, Oct 13 2012 */

%o (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 */

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(-x-x^2/2) ))); // _G. C. Greubel_, Sep 03 2023

%o (SageMath)

%o def A001464_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( exp(-x-x^2/2) ).egf_to_ogf().list()

%o A001464_list(40) # _G. C. Greubel_, Sep 03 2023

%Y Cf. A000085, A000178, A001147, A066325, A069834, A099174, A200675.

%Y Cf. A252284, A369755, A369756.

%K sign,easy

%O 0,4

%A _N. J. A. Sloane_, _J. H. Conway_ and _Simon Plouffe_

%E a(12) and a(13) corrected by _Simon Plouffe_