A096713 - OEIS (original) (raw)

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

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

The formulas about the p.d.f. of the standard normal distribution were proved, for example, by Charlier (1905, pp. 13-15), but they were well-known for many years before him. Charlier (1905) has generalized these results to other measures whose n-th moment (around 0) exists for each integer n >= 0.

Different forms (with or without signs) of these coefficients T(n,k) appear in other arrays as well; e.g., see A049403, A104556, A122848, A130757 (odd rows only), etc.

(End)

REFERENCES

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

FORMULA

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

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

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

Let He_n(x) be the n-th modified Hermite polynomial (see the references above); i.e., let He_n(x) = Sum_{k = 0..m} T(2*m, k)*x^(2*k) when n = 2*m and He_n(x) = Sum_{k = 0..m} T(2*m+1, k)*x^(2*k+1) when n = 2*m+1.

Let phi(x) = (1/sqrt(2*Pi)) * exp(-x^2/2) be the p.d.f. of a standard normal distribution. Then He_n(x) = (-1)^n * (1/phi(x)) * d^n(phi(x))/dx^n for n >= 0.

We have He_n(x) = x*He_{n-1}(x) - (n-1)*He_{n-2}(x) for n >= 2. (End)

EXAMPLE

Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:

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;

...

The corresponding modified Hermite polynomials are as follows

He_0(x) = 1, He_1(x) = x,

He_2(x) = -1 + x^2, He_3(x) = -3*x + x^3,

He_4(x) = 3 - 6*x^2 + x^4, He_5(x) = 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:

# Alternatively:

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

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

R. = ZZ[]

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

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

return T[::-1]

(Sage) # uses[bell_transform from 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

(Python)

from sympy import hermite, Poly, sqrt

def a(n): return Poly(hermite(n, x/sqrt(2))/2**(n/2), x).coeffs()[::-1]

for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017