A157280 a(n) arises in the normal ordering of n-th power of the operator (d/dx)(x(d/dx))^4.

1, 52, 43833, 149670844, 1346634725665, 25571928251231076, 893591647147188285577, 52327970757667659912764908, 4796836032234830356783078467969

Offset: 1

Author

Karol A. Penson, Feb 26 2009

Keywords

• nonn

Comments

this sequence generates the fifth terms of the following sequences:

a(2)=52=A000110(5), a(3)=43833=A020556(5), a(4)=149670844=A069223(5),

a(5)=1346634725665=A071379(5),a(6)=25571928251231076=A070227(5)

Programs

Mathematica

nMax = 8; kMax = 50; seq0 = {}; seq = {1}; While[seq != seq0, seq0 = seq; seq = (1/E Sum[HypergeometricPFQ[{k+1, k+1, k+1, k+1}, {1, 1, 1, 1}, x]/k!, {k, 0, kMax}] + O[x]^(nMax+1) // CoefficientList[#, x]&) Range[0, nMax]!^5 // Round; kMax += 10; Print[kMax]]; A157280 = seq (* Jean-François Alcover, Nov 07 2016 *)

Formula

Sequence defined through the following hypergeometric-type generating function, in Maple notation:

exp(-1)*sum(hypergeom([k+1,k+1,k+1,k+1],[1,1,1,1],x)/k!,k=0..infinity)=sum(a(n)*x^n/(n!)^5,n=0..infinity),

which is itself an infinite sum of hypergeometric functions.