A052782 a(n) = (5*n+1)^(n-1).

1, 1, 11, 256, 9261, 456976, 28629151, 2176782336, 194754273881, 20047612231936, 2334165173090451, 303305489096114176, 43513917611435838661, 6831675453247426400256, 1165087474585497590531111, 214481724045177216015794176, 42391158275216203514294433201

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..313
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 739
• J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Crossrefs

Cf. A000169, A052750, A052752, A052774, A000272, A001721.

Programs

Magma

[(5*n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017

Maple

spec := [S,{S=Set(B),B=Prod(Z,S,S,S,S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-5*x]/5], {x, 0, nmax}], x]*Range[0, nmax]!] (* or *) Table[(5*n+1)^(n-1), {n,0,50}] (* G. C. Greubel, Nov 16 2017 *)

PARI

for(n=0,50, print1((5*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017

PARI

x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-5*x)/5))) \\ G. C. Greubel, Nov 16 2017

Formula

E.g.f.: exp(-1/5*LambertW(-5*x)).

From Peter Bala, Dec 19 2013: (Start)

The e.g.f. A(x) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + ... satisfies:

1) A(x*exp(-5*x)) = exp(x) = 1/A(-x*exp(5*x));

2) A^5(x) = 1/x*series reversion(x*exp(-5*x));

3) A(x^5) = 1/x*series reversion(x*exp(-x^5));

4) A(x) = exp(x*A(x)^5);

5) A(x) = 1/A(-x*A(x)^10). (End)

E.g.f.: (-LambertW(-5*x)/(5*x))^(1/5). - Vaclav Kotesovec, Dec 07 2014

Related to A001721 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^5*log(1 + x) ) = series reversion(x - 11*x^2/2! + 107*x^3/3! - 1066*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016

Extensions

Better description from Vladeta Jovovic, Sep 02 2003