A100101 Bell(2n)*(2n-1)!!, where Bell are the Bell numbers A000110.

1, 2, 45, 3045, 434700, 109596375, 43800340815, 25797179878470, 21243510135522675, 23503974546075598575, 33865310276598741840900, 61964234361152712204340725, 141027420945032510510113517025

Offset

0,2

Comments

This sequence arises in the normal ordering problem the exponential of square of boson number operator.

Links

Table of n, a(n) for n=0..12.

Formula

a(n) = Bell(2*n)*(2*n)!/(2^n*n!) = A001147(n)*A000110(2*n).

E.g.f.: G(x) = Sum_{k>=0} exp((k*x)^2/2-1)/k!; a(n) = subs(x=0, (d^(2n)/dx^(2n))G(x)).

Mathematica

Array[BellB[2 #] (2 # - 1)!! &, 13, 0] (* Michael De Vlieger, Dec 24 2017 *)

Prog

(PARI) a(n)=round(exp(-1)*suminf(k=0, k^(2*n)/k!))*(2*n)!/(2^n*n!) \\ Charles R Greathouse IV, Nov 06 2011

Crossrefs

Cf. A000110, A001147.

Sequence in context: A092654 A374863 A209606 * A332244 A090601 A266016

Adjacent sequences: A100098 A100099 A100100 * A100102 A100103 A100104

Keyword

nonn

Author

Karol A. Penson, Nov 03 2004

Status

approved