login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A066325 Coefficients of unitary Hermite polynomials He_n(x). 7
1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 3, 0, -6, 0, 1, 0, 15, 0, -10, 0, 1, -15, 0, 45, 0, -15, 0, 1, 0, -105, 0, 105, 0, -21, 0, 1, 105, 0, -420, 0, 210, 0, -28, 0, 1, 0, 945, 0, -1260, 0, 378, 0, -36, 0, 1, -945, 0, 4725, 0, -3150, 0, 630, 0, -45, 0, 1, 0, -10395, 0, 17325, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Also number of involutions on n labeled elements with k fixed points times (-1)^(number of 2-cycles).

Also called normalized Hermite polynomials.

REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pg 89,94 (2.3.41,54).

LINKS

Table of n, a(n) for n=0..70.

P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6), Research Paper #R2.

E. Elizalde, Cosmology: techniques and observations, arXiv:gr-qc/0409076, 2004.

D. Foata, Une méthode combinatoire pour l'étude des fonctions spéciales

Index entries for sequences related to Hermite polynomials

FORMULA

T(n, k)=(-2)^((k-n)/2)*n!/(k!*((n-k)/2)!). n-k even. 0 otherwise.

E.g.f. (relative to x): A(x, y) = exp(x*y-x^2/2).

The umbral compositional inverses (Cf. A001147) of the polynomials He(n,x) are given by the same polynomials unsigned, A099174 . - Tom Copeland, Nov 15 2014

EXAMPLE

1;

0,  1;

-1, 0,  1;

0, -3,  0, 1;

3,  0, -6, 0, 1;

...

PROG

(Sage)

def A066325_row(n):

    T = [0]*(n+1)

    if n==1: return [1]

    for m in (1..n-1):

        a, b, c = 1, 0, 0

        for k in range(m, -1, -1):

            r = a - (k+1)*c

            if k < m : T[k+2] = u;

            a, b, c = T[k-1], a, b

            u = r

        T[1] = u;

    return T[1:]

for n in (1..11): A066325_row(n)  # Peter Luschny, Nov 01 2012

CROSSREFS

Row sums: A001464 (with different signs). Row sums of absolute values: A000085.

Cf. A060281.

Absolute values are given in A099174. - M. F. Hasler, Oct 08 2012

Cf. A001147.

Sequence in context: A126595 A247622 A179898 * A099174 A137297 A178117

Adjacent sequences:  A066322 A066323 A066324 * A066326 A066327 A066328

KEYWORD

sign,tabl

AUTHOR

Christian G. Bower, Dec 14 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 18 17:15 EST 2014. Contains 252173 sequences.