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A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.
(Formerly M3732 N1526)
7
1, -1, -1, 5, 1, -41, 31, 461, -895, -6481, 22591, 107029, -604031, -1964665, 17669471, 37341149, -567425279, -627491489, 19919950975, 2669742629, -759627879679, 652838174519, 31251532771999, -59976412450835, -1377594095061119, 4256461892701199, 64623242860354751 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Binomial transform gives A067994. Inverse binomial transform gives A062267(n)*(-1)^n. - Vladimir Reshetnikov, Oct 11 2016

REFERENCES

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 209.

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

T. D. Noe, Table of n, a(n) for n = 0..200

Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.

Index entries for sequences related to Hermite polynomials

FORMULA

E.g.f.: exp(-x-x^2).

a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*k!*C(n, k)*C(n-k, k).

a(n) = -a(n-1)-2*(n-1)*a(n-2), a(0) = 1, a(1) = -1.

A000186(n) ~ n!^2*exp(1)^(-3)*(a(0) + a(1)/n + a(2)/(2*[n]_2) + ... + a(k)/(k!*[n]_k) + ...), where [n]_k = n*(n-1)*...*(n-k + 1), [n]_0 = 1. - Vladeta Jovovic, Apr 30 2001

a(n) = sum{k=0..n, (-1)^(2*n-k)*C(k,n-k)*n!/k!}. - Paul Barry, Oct 08 2007, corrected by Altug Alkan, Oct 22 2015

Conjecture: a(n) +a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012

E.g.f.: 1 - x*(1 - E(0) )/(1+x) where E(k) = 1 - (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013

E.g.f.: -x/Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013

G.f.: 1/(x*Q(0)), where Q(k) = 1 + 1/x + 2*(k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013

MATHEMATICA

Table[HermiteH[n, -1/2], {n, 0, 25}] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)

PROG

(PARI)

N=66;  x='x+O('x^N);

egf=exp(-x-x^2);  Vec(serlaplace(egf))

/* Joerg Arndt, Mar 07 2013 */

(PARI) vector(50, n, n--; sum(k=0, n/2, (-1)^(n-k)*k!*binomial(n, k)*binomial(n-k, k))) \\ Altug Alkan, Oct 22 2015

(PARI) a(n) = polhermite(n, -1/2); \\ Michel Marcus, Oct 12 2016

(Python)

from sympy import hermite

def a(n): return hermite(n, -1/2) # Indranil Ghosh, May 26 2017

CROSSREFS

Cf. A000186, A062267, A144141.

Sequence in context: A158820 A082437 A039817 * A039922 A192353 A255979

Adjacent sequences:  A000318 A000319 A000320 * A000322 A000323 A000324

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Formulae and more terms from Vladeta Jovovic, Apr 30 2001

STATUS

approved

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Last modified June 28 04:43 EDT 2017. Contains 288813 sequences.