A271049 a(n) = Sum_{k>=0} (Gamma(3*n+k-1)*Gamma((k+2)/3)/(Gamma(k)*Gamma(k+1)*Gamma(n-1/3+k/3)))/(3^(n-1)*exp(1)). Dobinski-type relation.

1, 3, 105, 9157, 1467989, 372555091, 136839757533, 68506049319485, 44775766291961897, 36988728433561712899, 37653691956186855176881, 46283247358178623165789813, 67556279347568889520823938365, 115470391901500605263068596360787

Offset: 0

Author

Karol A. Penson, Mar 29 2016

Keywords

• nonn

Comments

Numbers appearing in the normal ordering of the n-th power of the differential operator x^3*(d/dx)*x^2*(d/dx). Generalized Bell numbers.

From Karol A. Penson, Apr 03 2016: (Start)

Integral representation as the n-th Stieltjes moment of a positive function W(x) on (0,infinity), in Maple notation:

a(n)=int(x^n*W(x),x=0..infinity),n=0,1... W(x) = Dirac(x)/exp(1) + BesselK(2/3, (2/3)*sqrt(x))*(-(1/9)*3^(1/6)*GAMMA(2/3)*hypergeom([], [4/3, 4/3, 5/3, 2], (1/243)*x)/(exp(1)*Pi*x^(1/3))-(2/3)*sqrt(3)*hypergeom([], [1/3, 2/3, 2/3, 4/3], (1/243)*x)/(exp(1)*Pi*x)-(2/9)*3^(1/3)*hypergeom([], [2/3, 1, 4/3, 5/3], (1/243)*x)/(exp(1)*x^(2/3)*GAMMA(2/3)))+BesselK(5/3, (2/3)*sqrt(x))*((1/18)*3^(1/6)*x^(1/6)*GAMMA(2/3)*hypergeom([], [4/3, 4/3, 5/3, 2], (1/243)*x)/(exp(1)*Pi)+(1/3)*sqrt(3)*hypergeom([], [1/3, 2/3, 2/3, 4/3], (1/243)*x)/(exp(1)*Pi*sqrt(x))+(1/9)*3^(1/3)*hypergeom([], [2/3, 1, 4/3, 5/3], (1/243)*x)/(exp(1)*x^(1/6)*GAMMA(2/3))), 0<=x<=infinity.

The function W(x) is everywhere positive, is singular at x=0 and it monotonically decreases everywhere when x>0, with limit(W(x),x=infinity) = 0. It contains a single Dirac delta function centered at x=0 and the continuous part expressed as a combination of two BesselK functions and six generalized hypergeometric functions of type 0F4. (End)

Programs

Maple

a:=proc(n)sum(GAMMA(3*n+k-1)*GAMMA((k+2)/3)/(GAMMA(k)*GAMMA(k+1)*GAMMA(n-1/3+k/3)),k=0..infinity)/(3^(n-1)*exp(1));end:
seq(a(n), n=0..10);

Formula

Special values of the hypergeometric functions of type 2F4, in Maple notation: a(n) = ((1/18)*GAMMA(3*n+2)*GAMMA(2/3)*hypergeom([n+1, n+4/3], [4/3, 4/3, 5/3, 2], 1/27)/GAMMA(n+2/3)+GAMMA(3*n)*hypergeom([n+2/3, n+1/3], [1/3, 2/3, 2/3, 4/3], 1/27)/GAMMA(n)+(1/9)*GAMMA(3*n+1)*Pi*sqrt(3)*hypergeom([n+1, n+2/3], [2/3, 1, 4/3, 5/3], 1/27)/(GAMMA(2/3)*GAMMA(n+1/3)))/(exp(1)*3^(n-1)), n=0,1,2... .