|
| |
|
|
A001464
|
|
E.g.f. exp( -x -(1/2)*x^2 ).
(Formerly M0361 N0137)
|
|
7
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Signed row sums of Hermite polynomials p(k, x) = x*p(k - 1, x) - (n - 1)*p(k - 2, x). - Roger Bagula, Oct 06 2006
|
|
|
REFERENCES
|
Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Books, New York, 1945, page 32
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
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
|
Table of n, a(n) for n=0..26.
|
|
|
FORMULA
|
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)*prod(i=0, k, 2*i-1)*x^(n-2*k)) - Benoit Cloitre, May 01 2003
a(n)=(-1)^(n+1)*sum(k=0, floor(n/2), (-1)^k*C(n, 2*k)*(2k-1)!!) - Benoit Cloitre, May 01 2003
a(0)=1, a(1)=-1; a(n)=-a(n-1)-(n-1)*a(n-2) - Matthew J. White (mattjameswhite(AT)hotmail.com), Mar 01 2006
G.f.: 1/(U(0) + x) where U(k)= 1 + x*(k+1) - x*(k+1)/(1 + x/U(k+1)) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 12 2012
G.f.: 1/U(0) where U(k)= 1 + x + x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Nov 04 2012
G.f.: 1/Q(0), where Q(k)= 1 + x*k + x/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013
|
|
|
MATHEMATICA
|
p[0, x] = 1; p[1, x] = x; p[k_, x_] := p[k, x] = x*p[k - 1, x] - (n - 1)*p[k - 2, x]; Table[Expand[p[n, x]], {n, 0, 10}]; Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[p[ n, x], x]]}], {n, 0, 15}]; - Roger Bagula, Oct 06 2006
With[{nn=30}, CoefficientList[Series[Exp[-x-1/2 x^2], {x, 0, nn}], x]Range[0, nn]!] (* From Harvey P. Dale, Sep 16 2011 *)
|
|
|
PROG
|
(PARI) Vec( serlaplace( exp( -x -(1/2)*x^2 + O(x^66) ) ) ) /* Joerg Arndt, Oct 13 2012 */
|
|
|
CROSSREFS
|
Cf. A099174, A000085, A066325.
Sequence in context: A071208 A216242 A083555 * 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
|
13th and 14-th terms corrected by Simon Plouffe
|
|
|
STATUS
|
approved
|
| |
|
|
