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A052774 a(n) = (4*n+1)^(n-1).

%I A052774 #36 Sep 08 2022 08:44:59
%S A052774 1,1,9,169,4913,194481,9765625,594823321,42618442977,3512479453921,
%T A052774 327381934393961,34050628916015625,3909821048582988049,
%U A052774 491258904256726154641,67046038752496061076057,9876832533361318095112441,1562069488955406402587890625
%N A052774 a(n) = (4*n+1)^(n-1).
%H A052774 G. C. Greubel, <a href="/A052774/b052774.txt">Table of n, a(n) for n = 0..320</a>
%H A052774 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=731">Encyclopedia of Combinatorial Structures 731</a>
%H A052774 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:1403.5962 [math.CO], 2014.
%F A052774 E.g.f.: exp(-1/4*LambertW(-4*x)).
%F A052774 a(n) = A016813(n)^A023443(n). - _Wesley Ivan Hurt_, Dec 03 2013
%F A052774 From _Peter Bala_, Dec 19 2013: (Start)
%F A052774 The e.g.f. A(x) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + ... satisfies:
%F A052774 1) A(x*exp(-4*x)) = exp(x) = 1/A(-x*exp(4*x));
%F A052774 2) A^4(x) = 1/x*series reversion(x*exp(-4*x));
%F A052774 3) A(x^4) = 1/x*series reversion(x*exp(-x^4));
%F A052774 4) A(x) = exp(x*A(x)^4);
%F A052774 5) A(x) = 1/A(-x*A(x)^8). (End)
%F A052774 E.g.f.: (-LambertW(-4*x)/(4*x))^(1/4). - _Vaclav Kotesovec_, Dec 07 2014
%F A052774 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
%p A052774 spec := [S,{B=Prod(Z,S,S,S,S),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052774 A052774:=n->(4*n+1)^(n-1); seq(A052774(n), n=0..20); # _Wesley Ivan Hurt_, Dec 03 2013
%t A052774 Table[(4n+1)^(n-1), {n,0,20}] (* _Wesley Ivan Hurt_, Dec 03 2013 *)
%t A052774 With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-4*x]/4], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 14 2017 *)
%o A052774 (PARI) for(n=0, 30, print1((4*n+1)^(n-1), ", ")) \\ _G. C. Greubel_, Nov 14 2017
%o A052774 (PARI) x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-4*x)/4))) \\ _G. C. Greubel_, Nov 14 2017
%o A052774 (Magma) [(4*n+1)^(n-1): n in [0..30]]; // _G. C. Greubel_, Nov 14 2017
%Y A052774 Cf. A000169, A052750, A052752, A052782, A000272, A001716.
%K A052774 nonn,easy
%O A052774 0,3
%A A052774 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052774 Better description from _Vladeta Jovovic_, Sep 02 2003