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%I #8 Sep 08 2022 08:45:43
%S 1,1,9,625,274625,671898241,8458700490625,520900360822838529,
%T 151632993506657159886849,203635581444958952230203985921,
%U 1239028497632876493535705227172341761
%N a(n) = (n*2^n + 1)^(n-1).
%C The g.f. is a special case (q=2) of the following identity.
%C Let W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! = LambertW(-x)/(-x), then
%C Sum_{n>=0} (n*q^n+1)^(n-1)/q^(n^2)*x^n/n! = Sum_{n>=0} W(x/q^n)^n/q^(n^2)*x^n/n!
%C where the radius of convergence is |x| <= q/e for q>=1.
%H G. C. Greubel, <a href="/A158881/b158881.txt">Table of n, a(n) for n = 0..55</a>
%F G.f.: A(x) = Sum_{n>=0} (n*2^n + 1)^(n-1)/2^(n^2) * x^n/n!
%F G.f.: A(x) = Sum_{n>=0} W(x/2^n)^n/2^(n^2) * x^n/n!, and
%F a(n)/2^(n^2) is the coefficient of x^n/n! in W(x)^(1/2^n)
%F where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
%F Radius of convergence of series A(x) is |x| <= 2/e.
%e G.f.: A(x) = 1 + 3^0/2*x + 9^1/2^4*x^2/2! + 25^2/2^9*x^3/3! + 65^3/2^16*x^4/4! + 161^4/2^25*x^5/5! +...
%e A(x) = 1 + W(x/2)/2*x + W(x/4)^2/2^4*x^2/2! + W(x/8)^3/2^9*x^3/3! +...
%e where W(x) = LambertW(-x)/(-x) so that W(x) = exp(x*W(x)).
%e Special values.
%e A(1/2) = 1.367881486725746399880346284881720747435653310931858829...
%e A(1/e) = 1.237164211886302867099485584025040050496738919299895839...
%e A(2/e) = 2.027079144901937613098735287853530386549370956336296669...
%e A(-2/e)= 0.733788551140988480682883862465033405661534959498406132...
%p seq( (n*2^n+1)^(n-1), n=0..10); # _G. C. Greubel_, Mar 04 2020
%t Table[(n*2^n+1)^(n-1),{n,0,10}] (* _Harvey P. Dale_, Jun 04 2015 *)
%o (PARI) a(n)=(n*2^n + 1)^(n-1);
%o (Magma) [(n*2^n+1)^(n-1): n in [0..10]]; // _G. C. Greubel_, Mar 04 2020
%o (Sage) [(n*2^n+1)^(n-1) for n in (0..10)] # _G. C. Greubel_, Mar 04 2020
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 22 2009
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