A052752 a(n) = (3*n+1)^(n-1).
1, 1, 7, 100, 2197, 65536, 2476099, 113379904, 6103515625, 377801998336, 26439622160671, 2064377754059776, 177917621779460413, 16777216000000000000, 1718264124282290785243, 189937030341242876870656, 22539340290692258087863249, 2857942574656970690381479936
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..333
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 708
- J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Programs
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Magma
[(3*n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017
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Maple
spec := [S,{B=Prod(S,S,S,Z),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20); -
Mathematica
Table[(3n+1)^(n-1),{n,0,20}] (* Harvey P. Dale, Aug 14 2015 *) With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-3*x]/3], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 16 2017 *) -
PARI
for(n=0,50, print1((3*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017
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PARI
x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-3*x)/3))) \\ G. C. Greubel, Nov 16 2017
Formula
E.g.f.: exp(-(1/3)*LambertW(-3*x)).
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + ... satisfies:
1) A(x*exp(-3*x)) = exp(x) = 1/A(-x*exp(3*x));
2) A^3(x) = 1/x*series reversion(x*exp(-3*x));
3) A(x^3) = 1/x*series reversion(x*exp(-x^3));
4) A(x) = exp(x*A(x)^3);
5) A(x) = 1/A(-x*A(x)^6). (End)
E.g.f.: (-LambertW(-3*x)/(3*x))^(1/3). - Vaclav Kotesovec, Dec 07 2014
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
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
Extensions
Better description from Vladeta Jovovic, Sep 02 2003