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%I M3732 N1526 #94 Apr 10 2023 10:52:08
%S 1,-1,-1,5,1,-41,31,461,-895,-6481,22591,107029,-604031,-1964665,
%T 17669471,37341149,-567425279,-627491489,19919950975,2669742629,
%U -759627879679,652838174519,31251532771999,-59976412450835,-1377594095061119,4256461892701199,64623242860354751
%N H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.
%C Binomial transform gives A067994. Inverse binomial transform gives A062267(n)*(-1)^n. - _Vladimir Reshetnikov_, Oct 11 2016
%C 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
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 209.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A000321/b000321.txt">Table of n, a(n) for n = 0..732</a> (terms 0..200 from T. D. Noe)
%H Koichi, Yamamoto, <a href="http://dx.doi.org/10.2969/jmsj/00140226">An asymptotic series for the number of three-line Latin rectangles</a>, J. Math. Soc. Japan 1, (1950). 226-241.
%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F E.g.f.: exp(-x-x^2).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*k!*C(n, k)*C(n-k, k).
%F a(n) = - a(n-1) - 2*(n-1)*a(n-2), a(0) = 1, a(1) = -1.
%F 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
%F 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
%F 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
%F 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
%F 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
%F 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
%F E.g.f.: Product_{k>=1} (1 + (-x)^k)^(mu(k)/k). - _Ilya Gutkovskiy_, May 26 2019
%t Table[HermiteH[n, -1/2], {n, 0, 25}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 15 2009 *)
%t Table[(-2)^n HypergeometricU[-n/2, 1/2, 1/4], {n, 0, 25}] (* _Benedict W. J. Irwin_, Oct 17 2017 *)
%o (PARI)
%o N=66; x='x+O('x^N);
%o egf=exp(-x-x^2); Vec(serlaplace(egf))
%o /* _Joerg Arndt_, Mar 07 2013 */
%o (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
%o (PARI) a(n) = polhermite(n, -1/2); \\ _Michel Marcus_, Oct 12 2016
%o (Python)
%o from sympy import hermite
%o def a(n): return hermite(n, -1/2) # _Indranil Ghosh_, May 26 2017
%o (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
%Y Cf. A000186, A062267, A144141, A293604.
%K sign,easy
%O 0,4
%A _N. J. A. Sloane_
%E Formulae and more terms from _Vladeta Jovovic_, Apr 30 2001
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