A157642 a(n) arises in the normal ordering of n-th power of the operator (d/dx)^3(x(d/dx))^3.

5, 1211, 1177071, 2851057633, 13702497878021, 114142446044738995, 1506157186706580123591, 29520950676106077642732801, 818899164643659779062878378373, 30959918834233822075763618062253451

Offset

1,1

Comments

Special values of a sum of three hypergeometric functions of type 3F5.

In Maple notation:

Links

Table of n, a(n) for n=1..10.

Formula

a(n)=exp(-1)*3^(3*n)*((1/6)*(n!)^3*hypergeom([n+1, n+1, n+1],

[1, 1, 4/3, 5/3, 2], 1/27)+(3/8)*sqrt(3)*GAMMA(2/3)^3*GAMMA(n+1/3)^3*hypergeom([n+1/3, n+1/3, n+1/3]

, [1/3, 1/3, 1/3, 2/3, 4/3], 1/27)/Pi^3

+(1/2)*GAMMA(n+2/3)^3*hypergeom([n+2/3, n+2/3, n+2/3], [2/3, 2/3, 2/3, 4/3, 5/3], 1/27)/GAMMA(2/3)^3),

, n=1,2... .

Mathematica

Round[Table[1/E*27^n*((3*Sqrt[3]*Gamma[2/3]^3*Gamma[1/3 + n]^3* HypergeometricPFQ[{1/3 + n, 1/3 + n, 1/3 + n}, {1/3, 1/3, 1/3, 2/3, 4/3}, 1/27])/(8*Pi^3) + 1/6*((3*Gamma[2/3 + n]^3* HypergeometricPFQ[{2/3 + n, 2/3 + n, 2/3 + n}, {2/3, 2/3, 2/3, 4/3, 5/3}, 1/27])/ Gamma[2/3]^3 + n!^3*HypergeometricPFQ[{1 + n, 1 + n, 1 + n}, {1, 1, 4/3, 5/3, 2}, 1/27])), {n, 1, 15}]] (* Vaclav Kotesovec, Jan 17 2026 *)

Crossrefs

Sequence in context: A317374 A189249 A034864 * A379987 A234811 A069642

Adjacent sequences: A157639 A157640 A157641 * A157643 A157644 A157645

Keyword

nonn

Author

Karol A. Penson, Mar 03 2009

Status

approved