A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n. (Formerly M3732 N1526)

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

Offset

0,4

Comments

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

The congruence a(n+k) == (-1)^k*a(n) (mod k) holds for all n and k. It follows that for even k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of k, while for odd k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of 2*k. See A047974. - Peter Bala, Apr 10 2023

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

Seiichi Manyama, Table of n, a(n) for n = 0..732 (terms 0..200 from T. D. Noe)

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

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

a(n) = (-2)^n * U(-n/2, 1/2, 1/4), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017

E.g.f.: Product_{k>=1} (1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019

Mathematica

Table[HermiteH[n, -1/2], {n, 0, 25}] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[(-2)^n HypergeometricU[-n/2, 1/2, 1/4], {n, 0, 25}] (* Benedict W. J. Irwin, Oct 17 2017 *)

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
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

Crossrefs

Cf. A000186, A062267, A144141, A293604.

Sequence in context: A308440 A039817 A293604 * A293573 A039922 A192353

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