A001464 Expansion of e.g.f. exp(-x - (1/2)*x^2). (Formerly M0361 N0137)

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

0,4

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

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

Cf. A252284, A369755, A369756.

Sequence in context: A083555 A233147 A336633 * A067136 A180068 A034439

Adjacent sequences: A001461 A001462 A001463 * A001465 A001466 A001467

Keyword

sign,easy

Author

N. J. A. Sloane, J. H. Conway and Simon Plouffe

Extensions

a(12) and a(13) corrected by Simon Plouffe

Status

approved