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

0,2

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

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)

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 *)

Crossrefs

Cf. A123510, A123511, A123512, A123525.

Sequence in context: A357421 A071798 A338514 * A122418 A069788 A283678

Adjacent sequences: A123683 A123684 A123685 * A123687 A123688 A123689

Keyword

nonn

Author

Karol A. Penson, Oct 06 2006

Extensions

More terms from Emeric Deutsch, Oct 31 2006

Status

approved