A123686 - OEIS

%I #11 May 10 2019 21:23:37

%S 1,2,54,2460,239190,33124140,6896500380,1879519201560,674900483206950,

%T 300426422192196300,164868151446145847700,108046627817926248851400,

%U 83890281074290204071858300,75722368306901033144261835000

%N E.g.f.: (1-x^4)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).

%C 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="/A123686/b123686.txt">Table of n, a(n) for n = 0..220</a>

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

%F Recurrence: n*a(n) = 2*(2*n - 1)*(3*n - 2)*a(n-1) + 4*(n-1)^2*(2*n - 3)*(2*n - 1)*(2*n + 1)*a(n-2) + 8*(n-2)^2*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 16*(n-3)^2*(n-2)^2*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4).

%F a(n) ~ 2^(2*n - 3/4) * exp(2*sqrt(2*n) - 2*n -1) * n^(2*n - 1/4) / sqrt(Pi) * (1 + 91/(48*sqrt(2*n))). (End)

%p G:=(1-x^4)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)): Gser:=series(G,x=0,40): seq((2*n)!*coeff(Gser,x,2*n),n=0..15); # _Emeric Deutsch_, Oct 31 2006

%t With[{nmax = 50}, CoefficientList[Series[(1 - x^4)^(-1/2)*Exp[x^2/(1 - x^2)]*BesselI[0, x^2/(x^2 - 1)], {x, 0, nmax}], x]*Range[0, nmax]!][[;; ;; 2 ]] (* _G. C. Greubel_, Oct 18 2017 *)

%Y Cf. A123510, A123511, A123512, A123525.

%K nonn

%O 0,2

%A _Karol A. Penson_, Oct 06 2006

%E More terms from _Emeric Deutsch_, Oct 31 2006