VOOZH about

URL: https://oeis.org/A052774

⇱ A052774 - OEIS

login
A052774
a(n) = (4*n+1)^(n-1).
6
1, 1, 9, 169, 4913, 194481, 9765625, 594823321, 42618442977, 3512479453921, 327381934393961, 34050628916015625, 3909821048582988049, 491258904256726154641, 67046038752496061076057, 9876832533361318095112441, 1562069488955406402587890625
OFFSET
0,3
LINKS
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
FORMULA
E.g.f.: exp(-1/4*LambertW(-4*x)).
a(n) = A016813(n)^A023443(n). - Wesley Ivan Hurt, Dec 03 2013
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + ... satisfies:
1) A(x*exp(-4*x)) = exp(x) = 1/A(-x*exp(4*x));
2) A^4(x) = 1/x*series reversion(x*exp(-4*x));
3) A(x^4) = 1/x*series reversion(x*exp(-x^4));
4) A(x) = exp(x*A(x)^4);
5) A(x) = 1/A(-x*A(x)^8). (End)
E.g.f.: (-LambertW(-4*x)/(4*x))^(1/4). - Vaclav Kotesovec, Dec 07 2014
Related to A001716 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^4*log(1 + x) ) = series reversion(x - 9*x^2/2! + 74*x^3/3! - 638*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016
MAPLE
spec := [S, {B=Prod(Z, S, S, S, S), S=Set(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
A052774:=n->(4*n+1)^(n-1); seq(A052774(n), n=0..20); # Wesley Ivan Hurt, Dec 03 2013
MATHEMATICA
Table[(4n+1)^(n-1), {n, 0, 20}] (* Wesley Ivan Hurt, Dec 03 2013 *)
With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-4*x]/4], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)
PROG
(PARI) for(n=0, 30, print1((4*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 14 2017
(PARI) x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-4*x)/4))) \\ G. C. Greubel, Nov 14 2017
(Magma) [(4*n+1)^(n-1): n in [0..30]]; // G. C. Greubel, Nov 14 2017
CROSSREFS
KEYWORD
nonn,easy
EXTENSIONS
Better description from Vladeta Jovovic, Sep 02 2003
STATUS
approved