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A096713 Triangle of nonzero coefficients of the modified Hermite polynomials. 8
1, 1, -1, 1, -3, 1, 3, -6, 1, 15, -10, 1, -15, 45, -15, 1, -105, 105, -21, 1, 105, -420, 210, -28, 1, 945, -1260, 378, -36, 1, -945, 4725, -3150, 630, -45, 1, -10395, 17325, -6930, 990, -55, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, 135135, -270270, 135135, -25740, 2145, -78, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Triangle of nonzero coefficients of matching polynomial of complete graph of order n.

Row sums of absolute values produce A000085 (number of involutions). - Wouter Meeussen, Mar 12 2008

Row n has floor(n/2)+1 nonzero coefficients. - Robert Israel, Dec 23 2015

Also the nonzero terms of the Bell matrix generated by the sequence [-1,1,0,0,0, ...] read by rows (see second Sage program). For the definition of the Bell matrix see A264428. - Peter Luschny, Jan 20 2016

REFERENCES

C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10099 (rows 0 to 199, flattened)

Eric Weisstein's World of Mathematics, Hermite Polynomial

Eric Weisstein's World of Mathematics, Matching Polynomial - Eric W. Weisstein, Sep 27 2008

FORMULA

G.f.: HermiteH(n,x/sqrt(2))/2^(n/2). - Wouter Meeussen, Mar 12 2008

From Robert Israel, Dec 23 2015: (Start)

T(2m, k) = (-1)^(m+k)*(2m)!*2^(k-m)/((m-k)!*(2k)!), k=0..m.

T(2m+1, k) = (-1)^(m+k)*(2m+1)!*2^(k-m)/((m-k)!*(2k+1)!), k=0..m. (End)

EXAMPLE

1, x, -1 + x^2, -3*x + x^3, 3 - 6*x^2 + x^4, 15*x - 10*x^3 + x^5, ...

MAPLE

A:= NULL:

for n from 0 to 20 do

  HH:= expand(orthopoly[H](n, x/sqrt(2))/2^(n/2));

  C:= subs(0=NULL, [seq(coeff(HH, x, j), j=0..n)]);

  A:= A, op(C);

od:

A; #  Robert Israel, Dec 23 2015

# Alternatively:

A096713 := (n, k) -> `if`(2*k<n, NULL, (-1/2)^(n-k)*n!/((2*k-n)!*(n-k)!)):

seq(seq(A096713(n, k), k=0..n), n=0..13); # Peter Luschny, Dec 24 2015

MATHEMATICA

Table[CoefficientList[HermiteH[n, x/Sqrt[2] ]/2^(n/2), x], {n, 0, 25}] (* Wouter Meeussen, Mar 12 2008 *)

PROG

(PARI) T(n, k)=if(k<0|2*k>n, 0, (-1)^(n\2-k)*n!/(n\2-k)!/(n%2+2*k)!/2^(n\2-k)) /* Michael Somos, Jun 04 2005 */

(Sage)

from sage.functions.hypergeometric import closed_form

def A096713_row(n):

    R.<z> = ZZ[]

    h = hypergeometric([-n/2, (1-n)/2], [], -2*z)

    T = R(closed_form(h)).coefficients()

    return T[::-1]

for n in range(13): A096713_row(n) # Peter Luschny, Aug 21 2014

(Sage) # Alternatively:

# The function bell_transform is defined in A264428.

def bell_zero_filter(generator, dim):

    G = [generator(k) for k in srange(dim)]

    row = lambda n: bell_transform(n, G)

    F = [filter(lambda r: r != 0, R) for R in [row(n) for n in srange(dim)]]

    return [i for f in F for i in f]

print bell_zero_filter(lambda n: [1, -1][n] if n < 2 else 0, 14) # Peter Luschny, Jan 20 2016

CROSSREFS

Cf. A000085.

Sequence in context: A055885 A181425 A174505 * A107726 A114159 A236560

Adjacent sequences:  A096710 A096711 A096712 * A096714 A096715 A096716

KEYWORD

sign,tabf

AUTHOR

Eric W. Weisstein, Jul 04 2004

STATUS

approved

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Last modified December 4 05:11 EST 2016. Contains 278748 sequences.