A123512 - OEIS

%I #22 Nov 13 2017 09:12:26

%S 1,10,105,1190,14630,194796,2798670,43204260,713655855,12564061510,

%T 234896893231,4648313235930,97068707038940,2133251854548920,

%U 49215687006553740,1189262114277026856,30037396074996304365

%N Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

%H G. C. Greubel, <a href="/A123512/b123512.txt">Table of n, a(n) for n = 0..439</a>

%F E.g.f.: (1/(1-x)^5)*exp(x/(1-x))*LaguerreL(4,-x/(1-x)).

%F From _Vaclav Kotesovec_, Nov 13 2017: (Start)

%F Recurrence: n*a(n) = 2*n*(n+4)*a(n-1) - (n-1)*(n+3)*(n+4)*a(n-2).

%F a(n) ~ exp(2*sqrt(n)-n-1/2) * n^(n + 17/4) / (3*2^(7/2)) * (1 + 31/(48*sqrt(n))).

%F (End)

%t CoefficientList[ Series[(1/(1 - x)^5)*Exp[x/(1 - x)]LaguerreL[4, -x/(1 - x)], {x,0,16}], x]*Range[0, 16]! (* _Robert G. Wilson v_, Oct 03 2006 *)

%o (PARI)

%o LaguerreL(n,v='x) = {

%o   my(x='x+O('x^(n+1)), t='t);

%o   subst(polcoeff(exp(-x*t/(1-x))/(1-x), n), 't, v);

%o };

%o N=17;x='x+O('x^N); Vec(serlaplace((1/(1-x)^5)*exp(x/(1-x))*LaguerreL(4,-x/(1-x)))) \\ _Gheorghe Coserea_, Oct 26 2017

%Y Cf.: A002720, A052852, A123510, A123511.

%K nonn

%O 0,2

%A _Karol A. Penson_, Oct 02 2006

%E a(0)=1 prepended by _Gheorghe Coserea_, Oct 26 2017