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A360547 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^2 ).

%I A360547 #23 Feb 16 2025 08:34:04
%S A360547 1,1,9,121,2417,64721,2180665,88719625,4233968737,231991022881,
%T A360547 14356691152361,990506937621785,75390334060230865,6275675303410022641,
%U A360547 567191776288882702105,55313848534122299876521,5789703106014903009828545,647414950001156861671249985
%N A360547 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^2 ).
%H A360547 Seiichi Manyama, <a href="/A360547/b360547.txt">Table of n, a(n) for n = 0..339</a>
%H A360547 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A360547 E.g.f.: A(x) = exp( (-1/2) * LambertW(-2*x * exp(2*x)) ).
%F A360547 E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(2*x)) / (2*x * exp(2*x)) ).
%F A360547 E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(2*x))^k / k! ).
%F A360547 a(n) = Sum_{k=0..n} (2*k)^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
%F A360547 a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n - 1/2) * LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Feb 17 2023
%t A360547 nmax = 20; A[_] = 1;
%t A360547 Do[A[x_] = Exp[x*(Exp[x]*A[x])^2] + O[x]^(nmax+1) // Normal, {nmax}];
%t A360547 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A360547 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(2*x))/2)))
%o A360547 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(2*x))/(2*x*exp(2*x)))))
%o A360547 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(2*x))^k/k!))))
%o A360547 (PARI) a(n) = sum(k=0, n, (2*k)^(n-k)*(2*k+1)^(k-1)*binomial(n, k));
%Y A360547 Cf. A273953, A273954, A360544.
%Y A360547 Cf. A360465, A360473.
%K A360547 nonn
%O A360547 0,3
%A A360547 _Seiichi Manyama_, Feb 11 2023