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Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).
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%I #19 Jan 27 2023 18:59:11

%S 1,2,9,50,361,3042,29929,331298,4100625,55777922,828691369,

%T 13316140818,230256982201,4257449540450,83834039024649,

%U 1750225301567618,38614608429012001,897325298084953602,21904718673762721225,560258287738117292018,14981472258320814527241

%N Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>.

%F E.g.f.: exp(2*x/(1-x))/sqrt(1-x^2).

%F a(n) = |H_n(i)|^2 / 2^n = H_n(i) * H_n(-i) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1).

%F D-finite with recurrence: (n+2)*(a(n) + n*a(n-1)) = a(n+1) + n*(n-1)^2*a(n-2).

%F a(n) ~ n^n / (2 * exp(1 - 2*sqrt(2*n) + n)) * (1 + 2*sqrt(2)/(3*sqrt(n))). - _Vaclav Kotesovec_, Oct 27 2021

%t Table[Abs[HermiteH[n, I]]^2/2^n, {n, 0, 20}]

%t With[{nn=20},CoefficientList[Series[Exp[2x/(1-x)]/Sqrt[1-x^2],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jan 27 2023 *)

%Y Cf. A000321, A000898, A059343, A062267, A067994, A277280, A277281.

%K nonn

%O 0,2

%A _Vladimir Reshetnikov_, Oct 11 2016