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A066325 Coefficients of unitary Hermite polynomials He_n(x). 5
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

E. Elizalde, Cosmology: techniques and observations

D. Foata, Une methode combinatoire pour l'\'{e}tude des fonctions sp\'{e}ciales

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)

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

Sequence in context: A035653 A126595 A179898 * A099174 A137297 A178117

Adjacent sequences:  A066322 A066323 A066324 * A066326 A066327 A066328

KEYWORD

sign,tabl

AUTHOR

Christian G. Bower, Dec 14 2001

STATUS

approved

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Last modified July 31 13:31 EDT 2014. Contains 245085 sequences.