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E.g.f. satisfies A(x) = exp(x * exp(2*x) * A(x)).
3

%I #16 Feb 08 2023 07:51:27

%S 1,1,7,64,829,14056,295399,7426252,217637305,7291538704,275050426411,

%T 11540336658676,533224609095061,26908386824872216,1472691380336896399,

%U 86892807951798473116,5498668489586321670769,371511527654280649783840

%N E.g.f. satisfies A(x) = exp(x * exp(2*x) * A(x)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: A(x) = exp( -LambertW(-x * exp(2*x)) ).

%F E.g.f.: A(x) = -LambertW(-x * exp(2*x)) / (x * exp(2*x)).

%F E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x * exp(2*x))^k / k!.

%F a(n) = Sum_{k=0..n} (2*k)^(n-k) * (k+1)^(k-1) * binomial(n,k).

%F a(n) ~ sqrt(1+LambertW(2*exp(-1))) * 2^n * n^(n-1) / (exp(n-1) * LambertW(2*exp(-1))^n). - _Vaclav Kotesovec_, Feb 08 2023

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(2*x)))))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-lambertw(-x*exp(2*x))/(x*exp(2*x))))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(2*x))^k/k!)))

%o (PARI) a(n) = sum(k=0, n, (2*k)^(n-k)*(k+1)^(k-1)*binomial(n, k));

%Y Cf. A038050, A273954, A357247, A360466.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 08 2023