cp's OEIS Frontend

A052752 a(n) = (3*n+1)^(n-1).

%I A052752 #47 Aug 20 2025 00:28:03
%S A052752 1,1,7,100,2197,65536,2476099,113379904,6103515625,377801998336,
%T A052752 26439622160671,2064377754059776,177917621779460413,
%U A052752 16777216000000000000,1718264124282290785243,189937030341242876870656,22539340290692258087863249,2857942574656970690381479936
%N A052752 a(n) = (3*n+1)^(n-1).
%H A052752 G. C. Greubel, <a href="/A052752/b052752.txt">Table of n, a(n) for n = 0..333</a>
%H A052752 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=708">Encyclopedia of Combinatorial Structures 708</a>
%H A052752 J.-C. Novelli and 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 A052752 E.g.f.: exp(-(1/3)*LambertW(-3*x)).
%F A052752 From _Peter Bala_, Dec 19 2013: (Start)
%F A052752 The e.g.f. A(x) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + ... satisfies:
%F A052752 1) A(x*exp(-3*x)) = exp(x) = 1/A(-x*exp(3*x));
%F A052752 2) A^3(x) = 1/x*series reversion(x*exp(-3*x));
%F A052752 3) A(x^3) = 1/x*series reversion(x*exp(-x^3));
%F A052752 4) A(x) = exp(x*A(x)^3);
%F A052752 5) A(x) = 1/A(-x*A(x)^6). (End)
%F A052752 E.g.f.: (-LambertW(-3*x)/(3*x))^(1/3). - _Vaclav Kotesovec_, Dec 07 2014
%F A052752 Related to A001711 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^3*log(1 + x) ) = series reversion(x - 7*x^2/2! + 47*x^3/3! - 342*x^4/4! + ...). Cf. A000272, A052750. - _Peter Bala_, Jun 15 2016
%F A052752 a(n) = Sum_{k=1..n} (-1)^(n-k)*(2n+k)^(n-1)*binomial(n,k-1), a(0)=1. - _Vladimir Kruchinin_, Aug 14 2025
%p A052752 spec := [S,{B=Prod(S,S,S,Z),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052752 Table[(3n+1)^(n-1),{n,0,20}] (* _Harvey P. Dale_, Aug 14 2015 *)
%t A052752 With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-3*x]/3], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 16 2017 *)
%o A052752 (PARI) for(n=0,50, print1((3*n+1)^(n-1), ", ")) \\ _G. C. Greubel_, Nov 16 2017
%o A052752 (PARI) x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-3*x)/3))) \\ _G. C. Greubel_, Nov 16 2017
%o A052752 (Magma) [(3*n+1)^(n-1): n in [0..50]]; // _G. C. Greubel_, Nov 16 2017
%Y A052752 Cf. A000169, A052750, A052774, A052782, A000272, A001711.
%K A052752 nonn,easy,changed
%O A052752 0,3
%A A052752 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052752 Better description from _Vladeta Jovovic_, Sep 02 2003