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%I #18 Mar 12 2024 02:40:33
%S 1,1,1,17,65,1457,10657,307841,3403521,121414689,1810995009,
%T 77157569073,1453708980033,72128854709329,1644987113677793,
%U 93235988902015009,2498605538747794433,159345372352540823361,4909485778021467744897,348042700926255242296657
%N E.g.f.: exp( Shw(x) ) where Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.
%H Vincenzo Librandi, <a href="/A216143/b216143.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: ( -LambertW(-x)*LambertW(x)/x^2 )^(1/(2*x)).
%F a(n) ~ c * n^(n-1), where c = -1/2*exp(1-exp(1))*(LambertW(exp(-1))^(-exp(1)/2)-LambertW(exp(-1))^(exp(1)/2)*exp(2*exp(1))) = 3.11403283... if n is even, and c = 1/2*exp(1-exp(1))*(LambertW(exp(-1))^(-exp(1)/2)+LambertW(exp(-1))^(exp(1)/2)*exp(2*exp(1))) = 4.13355253... if n is odd. - _Vaclav Kotesovec_, Jul 08 2013
%e E.g.f.: A(x) = 1 + x + x^2/2! + 17*x^3/3! + 65*x^4/4! + 1457*x^5/5! +...
%e such that A(x) = exp( Shw(x) ) where
%e Shw(x) = x + 16*x^3/3! + 1296*x^5/5! + 262144*x^7/7! + 100000000*x^9/9! +...+ (2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! +...
%e Related expansions:
%e Chw(x) = 1 + 3*x^2/2! + 125*x^4/4! + 16807*x^6/6! + 4782969*x^8/8! +...+ (2*n+1)^(2*n-1)*x^(2*n)/(2*n)! +...
%e where Chw(x) + Shw(x) = LambertW(-x)/(-x).
%e Also, A(x)^x is an even function:
%e A(x)^x = 1 + 2*x^2/2! + 76*x^4/4! + 9816*x^6/6! + 2731408*x^8/8! +...+ A215880(2*n)*x^(2*n)/(2*n)! +....
%t max = 19; Shw[x_] := -(ProductLog[-x] + ProductLog[x])/(2*x); se = Series[Exp[Shw[x]] , {x, 0, max}]; (CoefficientList[se, x] // DeleteCases[#, 0] &)*Range[0, max]! (* _Jean-François Alcover_, Jun 24 2013 *)
%o (PARI) {a(n)=n!*polcoeff(exp(sum(m=0, n,(2*m+2)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)), n)}
%o (PARI) {a(n)=local(LW=sum(m=1, n+2,-(-1)^m*m^(m-1)*x^m/m!)+x^4*O(x^n)); n!*polcoeff( (-subst(LW, x, -x)*LW/x^2)^(1/(2*x)), n)}
%o for(n=0,21,print1(a(n),", "))
%Y Cf. A215880, A215881.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 02 2012
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