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

%I #33 Sep 08 2022 08:44:59

%S 1,1,11,256,9261,456976,28629151,2176782336,194754273881,

%T 20047612231936,2334165173090451,303305489096114176,

%U 43513917611435838661,6831675453247426400256,1165087474585497590531111,214481724045177216015794176,42391158275216203514294433201

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

%H G. C. Greubel, <a href="/A052782/b052782.txt">Table of n, a(n) for n = 0..313</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=739">Encyclopedia of Combinatorial Structures 739</a>

%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

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

%F From _Peter Bala_, Dec 19 2013: (Start)

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

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

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

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

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

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

%F E.g.f.: (-LambertW(-5*x)/(5*x))^(1/5). - _Vaclav Kotesovec_, Dec 07 2014

%F 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

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

%t 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 *)

%o (PARI) for(n=0,50, print1((5*n+1)^(n-1), ", ")) \\ _G. C. Greubel_, Nov 16 2017

%o (PARI) x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-5*x)/5))) \\ _G. C. Greubel_, Nov 16 2017

%o (Magma) [(5*n+1)^(n-1): n in [0..50]]; // _G. C. Greubel_, Nov 16 2017

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

%K nonn,easy

%O 0,3

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

%E Better description from _Vladeta Jovovic_, Sep 02 2003