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%I #12 May 11 2018 21:09:11
%S 1,3,13,63,345,2043,13029,88119,629169,4707315,36772029,298608687,
%T 2513795337,21874602987,196341166485,1814001266727,17222473789281,
%U 167763502438371,1674418724986221,17102228350521375,178562508150516921,1903865792493260763
%N E.g.f.: exp(3*x + 2*x^2).
%H Michael De Vlieger, <a href="/A202837/b202837.txt">Table of n, a(n) for n = 0..658</a>
%H Magdalena Boos, Giovanni Cerulli Irelli, Francesco Esposito, <a href="https://arxiv.org/abs/1802.06425">Parabolic orbits of 2-nilpotent elements for classical groups</a>, arXiv:1802.06425 [math.RT], 2018.
%F a(n) = Sum_{k=0..[n/2]} 3^(n-2*k)*2^k * n!/((n-2*k)!*k!).
%F O.g.f.: 1/(1-3*x - 4*x^2/(1-3*x - 8*x^2/(1-3*x - 12*x^2/(1-3*x - 16*x^2/(1-3*x -...))))), a continued fraction.
%F Recurrence: a(n) = 3*a(n-1) + 4*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ 2^(n-1/2)*exp(3/2*sqrt(n)-n/2-9/16)*n^(n/2)*(1+33/(64*sqrt(n))). - _Vaclav Kotesovec_, Oct 20 2012
%e E.g.f.: 1 + 3*x + 13*x^2/2! + 63*x^3/3! + 345*x^4/4! + 2043*x^5/5! +...
%t CoefficientList[Series[E^(3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o (PARI) {a(n)=n!*polcoeff(exp(3*x+2*x^2+x*O(x^n)),n)}
%o (PARI) {a(n)=sum(k=0,n\2,3^(n-2*k)*2^k*n!/((n-2*k)!*k!))}
%o (PARI) /* O.g.f. as a continued fraction: */
%o {a(n)=local(CF=1+3*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-3*x-4*(n-k)*x^2*CF)); polcoeff(CF, n)}
%Y Cf. A202836.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 25 2011
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