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A013703 Series(W(exp(1)*(1+x)), x) = sum( a[ n ]/(2^(2*n)*n!), n=0..infinity), where W is the Lambert W function.

%I A013703 #33 Nov 24 2024 14:00:42
%S A013703 1,2,-6,38,-370,4874,-81046,1628710,-38393538,1038795658,-31730277062,
%T A013703 1080038539942,-40538501660306,1663428036271754,-74080097240364918,
%U A013703 3558651343664651174,-183423140013051563746,10097324775041880827402,-591270189493633774009510
%N A013703 Series(W(exp(1)*(1+x)), x) = sum( a[ n ]/(2^(2*n)*n!), n=0..infinity), where W is the Lambert W function.
%C A013703 The n-th derivative of W(x) at x=exp(1) is exp(-n) * a(n) / 2^(2*n). - _Paolo Bonzini_, Jun 23 2016
%H A013703 Alois P. Heinz, <a href="/A013703/b013703.txt">Table of n, a(n) for n = 0..150</a>
%F A013703 E.g.f.: LambertW(exp(1)*(1+4*x)). - _Vladeta Jovovic_, Nov 19 2003
%F A013703 |a(n)| ~ 4^n * exp(n) * n^(n-1) / (1+exp(2))^(n-1/2). - _Vaclav Kotesovec_, Jul 09 2013
%e A013703 LambertW(exp(1)*(1+4*x)) = 1 + 1/2*x - 3/16*x^2 + 19/192*x^3  - 185/3072*x^4 + ...
%p A013703 LambertW( exp(1)*(1+x) );
%p A013703 seq(n!*coeff(series(LambertW(exp(1)*(1+4*x)), x, n+1), x, n), n=0..20); # _Vaclav Kotesovec_, Jul 09 2013
%t A013703 max = 17; (Series[ ProductLog[E*(1 + 4*x)], {x, 0, max}] // CoefficientList[#, x] &)*Range[0, max]! (* _Jean-François Alcover_, Jun 20 2013 *)
%o A013703 (Maxima)
%o A013703 a(n):= if n<2 then n+1 else 2*(n-1)!*(sum((binomial(n+k-1, n-1)*sum(binomial(k, j)*2^(n-j-1)*sum(binomial(j, l)*(-1)^(l)*sum((l^(n+j-i-1)*binomial(l, i))/(n+j-i-1)!, i, 0, l), l, 1, j), j, 1, k)), k, 1, n-1)); /* based on A042977, _Paolo Bonzini_, Jun 23 2016 */
%o A013703 (PARI) x='x+O('x^50); Vec(serlaplace(lambertw(exp(1)*(1+4*x)))) \\ _G. C. Greubel_, Nov 15 2017
%Y A013703 Twice row sums of A042977.
%K A013703 sign
%O A013703 0,2
%A A013703 Robert Corless (rmc(AT)pineapple.apmaths.uwo.ca)
%E A013703 More terms from _N. J. A. Sloane_.
%E A013703 More terms from _Vincenzo Librandi_, Jul 25 2013