OFFSET
0,4
COMMENTS
From Robert Israel, Apr 27 2017: (Start)
(-1)^n*a(n) is (the number of even involutions) - (the number of odd involutions) in the symmetric group S_n.
a(n) == (-1)^n (mod A069834(n-1)) for n >= 3.
a(n) is divisible by n-2 and by A200675(n+2). (End)
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.
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
EXAMPLE
G.f. = 1 - x + 2*x^3 - 2*x^4 - 6*x^5 + 16*x^6 + 20*x^7 - 132*x^8 + ...
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 *)
PROG
(PARI) Vec( serlaplace( exp( -x -(1/2)*x^2 + O(x^66) ) ) ) /* Joerg Arndt, Oct 13 2012 */
(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 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(-x-x^2/2) ))); // G. C. Greubel, Sep 03 2023
(SageMath)
def A001464_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(-x-x^2/2) ).egf_to_ogf().list()
A001464_list(40) # G. C. Greubel, Sep 03 2023
CROSSREFS
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
a(12) and a(13) corrected by Simon Plouffe
STATUS
approved