A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.
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
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
Programs
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Magma
m:=30; R
:=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 -
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 *) -
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
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PARI
a(n) = polhermite(n, -1/2); \\ Michel Marcus, Oct 12 2016
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Python
from sympy import hermite def a(n): return hermite(n, -1/2) # Indranil Ghosh, May 26 2017
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
Extensions
Formulae and more terms from Vladeta Jovovic, Apr 30 2001
Comments