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A360544 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(3/2) ).

%I A360544 #25 Feb 16 2025 08:34:04
%S A360544 1,1,7,73,1117,22741,580159,17826985,641494249,26473635865,
%T A360544 1232945359111,63978649829161,3660871368065509,229016870623703917,
%U A360544 15550838554432967647,1139139301403727884521,89544381521098908259729,7518611017848248249471089
%N A360544 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(3/2) ).
%H A360544 Seiichi Manyama, <a href="/A360544/b360544.txt">Table of n, a(n) for n = 0..352</a>
%H A360544 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A360544 E.g.f.: A(x) = exp( (-2/3) * LambertW(-3*x/2 * exp(3*x/2)) ).
%F A360544 E.g.f.: A(x) = ( -LambertW(-3*x/2 * exp(3*x/2)) / (3*x/2 * exp(3*x/2)) )^(2/3).
%F A360544 E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (3*x/2 * exp(3*x/2))^k / k! )^(2/3).
%F A360544 a(n) = (1/2^(n-1)) * Sum_{k=0..n} (3*k)^(n-k) * (3*k+2)^(k-1) * binomial(n,k).
%F A360544 a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n - 2/3) * LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Feb 17 2023
%o A360544 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-3/2*x*exp(3*x/2))/3)))
%o A360544 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2*exp(3*x/2)))^(2/3)))
%o A360544 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((sum(k=0, N, (k+1)^(k-1)*(3*x/2*exp(3*x/2))^k/k!))^(2/3)))
%o A360544 (PARI) a(n) = sum(k=0, n, (3*k)^(n-k)*(3*k+2)^(k-1)*binomial(n, k))/2^(n-1);
%Y A360544 Cf. A273953, A273954, A360547.
%Y A360544 Cf. A360473, A360545.
%K A360544 nonn
%O A360544 0,3
%A A360544 _Seiichi Manyama_, Feb 11 2023