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A360473 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^2 ).

%I A360473 #27 Feb 16 2025 08:34:04
%S A360473 1,1,7,82,1441,34036,1013149,36446698,1538703457,74607811048,
%T A360473 4086635087701,249593193648646,16819085803158577,1239637405609740268,
%U A360473 99206330021667838285,8567230421555333516746,794104205843228382969409
%N A360473 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^2 ).
%H A360473 Seiichi Manyama, <a href="/A360473/b360473.txt">Table of n, a(n) for n = 0..344</a>
%H A360473 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A360473 a(n) = Sum_{k=0..n} k^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
%F A360473 E.g.f.: A(x) = exp( -LambertW(-2*x * exp(x))/2 ).
%F A360473 E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(x)) / (2*x * exp(x)) ).
%F A360473 E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(x))^k / k! ).
%F A360473 a(n) ~ sqrt(1 + LambertW(exp(-1)/2)) * n^(n-1) / (2 * exp(n - 1/2) * LambertW(exp(-1)/2)^n). - _Vaclav Kotesovec_, Feb 17 2023
%t A360473 nmax = 20; A[_] = 1;
%t A360473 Do[A[x_] = Exp[x*Exp[x]*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
%t A360473 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A360473 (PARI) a(n) = sum(k=0, n, k^(n-k)*(2*k+1)^(k-1)*binomial(n, k));
%o A360473 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(x))/2)))
%o A360473 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(x))/(2*x*exp(x)))))
%o A360473 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(x))^k/k!))))
%Y A360473 Cf. A273954, A360465, A360474.
%K A360473 nonn
%O A360473 0,3
%A A360473 _Seiichi Manyama_, Feb 08 2023