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%I #40 Sep 08 2022 08:45:03
%S 1,2,2,-4,-20,-8,184,464,-1648,-10720,8224,230848,280768,-4978816,
%T -17257600,104891648,727511296,-1901510144,-28538404352,11377556480,
%U 1107214478336,1759326697472,-42984354695168,-163379084079104
%N Row sums of (signed) triangle A060821 (Hermite polynomials).
%H G. C. Greubel, <a href="/A062267/b062267.txt">Table of n, a(n) for n = 0..730</a>
%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F a(n) = Sum_{m=0..n} A060821(n, m) = H(n, 1), with the Hermite polynomials H(n, x).
%F E.g.f.: exp(-x*(x-2)).
%F a(n) = 2*(a(n - 1) - (n - 1)*a(n - 2)). - _Roger L. Bagula_, Sep 11 2006
%F a(n) = 2^n * U(-n/2, 1/2, 1), where U is the confluent hypergeometric function. - _Benedict W. J. Irwin_, Oct 17 2017
%F E.g.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(mu(k)/k). - _Ilya Gutkovskiy_, May 26 2019
%p A062267 := proc(n)
%p HermiteH(n,1) ;
%p simplify(%) ;
%p end proc: # _R. J. Mathar_, Feb 05 2013
%t lst={};Do[p=HermiteH[n,1];AppendTo[lst,p],{n,0,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jun 15 2009 *)
%t Table[2^n HypergeometricU[-n/2, 1/2, 1], {n, 0, 23}] (* _Benedict W. J. Irwin_, Oct 17 2017 *)
%t With[{nmax=50}, CoefficientList[Series[Exp[x*(2-x)], {x,0,nmax}],x]* Range[0, nmax]!] (* _G. C. Greubel_, Jun 08 2018 *)
%o (Python)
%o from sympy import hermite, Poly
%o def a(n): return sum(Poly(hermite(n, x), x).all_coeffs()) # _Indranil Ghosh_, May 26 2017
%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(-x*(x-2)))) \\ _G. C. Greubel_, Jun 08 2018
%o (PARI) a(n) = polhermite(n,1); \\ _Michel Marcus_, Jun 09 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(2-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jun 08 2018
%Y Cf. A000898, A121966.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Jun 19 2001
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