A123686 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).

1, 2, 54, 2460, 239190, 33124140, 6896500380, 1879519201560, 674900483206950, 300426422192196300, 164868151446145847700, 108046627817926248851400, 83890281074290204071858300, 75722368306901033144261835000

Offset: 0

Author

Karol A. Penson, Oct 06 2006

Keywords

• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..220

Crossrefs

Cf. A123510, A123511, A123512, A123525.

Programs

Maple

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

Mathematica

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 *)

Formula

From Vaclav Kotesovec, Nov 13 2017: (Start)

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).

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)

Extensions

More terms from Emeric Deutsch, Oct 31 2006