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%I #42 Sep 08 2022 08:45:11
%S 1,3,14,86,648,5752,58576,671568,8546432,119401856,1815177984,
%T 29808908032,525586164736,9898343691264,198227905206272,
%U 4204989697906688,94163381359509504,2219240984918720512,54898699229094412288,1422015190821016633344,38484192401958599131136
%N Exponential generating function is exp(2*x/(1-x))/(1-x).
%H Vincenzo Librandi, <a href="/A087912/b087912.txt">Table of n, a(n) for n = 0..200</a>
%H OEIS Wiki, <a href="https://oeis.org/wiki/Generating_functions#Exponential_generating_functions">Exponential generating functions</a>.
%F E.g.f.: exp(2*x/(1-x))/(1-x). - _M. F. Hasler_, Sep 30 2012
%F a(n) = n!*LaguerreL(n, -2).
%F a(n) = Sum_{k=0..n} 2^k*(n-k)!*binomial(n, k)^2.
%F E.g.f.: exp(x) * Sum_{n>=0} 2^n*x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [_Paul D. Hanna_, Nov 18 2011]
%F a(n) ~ n^(n+1/4)*exp(2*sqrt(2*n)-n-1)*2^(-3/4). - _Vaclav Kotesovec_, Sep 29 2012
%F Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(2*n))) = exp(-1). - _Vaclav Kotesovec_, Oct 12 2016
%F a(n) = n! * A160615(n)/A160616(n). - _Alois P. Heinz_, Jun 28 2017
%F D-finite with recurrence: a(n) +(-2*n-1)*a(n-1) +(n-1)^2*a(n-2)=0. - _R. J. Mathar_, Feb 21 2020
%p a := proc(n) option remember: if n<1 then 1 else (2*n+1)*a(n-1) - (n-1)^2*a(n-2) fi end: 'a(n)'$n=0..17; # _Zerinvary Lajos_, Sep 26 2006; corrected by _M. F. Hasler_, Sep 30 2012
%t Table[n! SeriesCoefficient[E^(2*x/(1-x))/(1-x), {x, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, May 10 2013 *)
%t Table[n!*LaguerreL[n, -2], {n, 0, 30}] (* _G. C. Greubel_, May 16 2018 *)
%o (PARI) A087912(n)={n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0,n,2^m*x^m/m!^2) ,n)} \\ _Paul D. Hanna_, Nov 18 2011
%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(2*x/(1-x))/(1-x))) \\ _Joerg Arndt_, May 10 2013
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 16 2018
%Y Cf. A002720, A274246, A277372.
%Y Column k=2 of A289192.
%Y Cf. A160615, A160616.
%K nonn
%O 0,2
%A _Vladeta Jovovic_, Oct 18 2003
%E Several minor edits by _M. F. Hasler_, Sep 30 2012
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