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A372278 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(5/2)) ).

%I A372278 #10 Feb 16 2025 08:34:06
%S A372278 1,2,14,218,5256,172332,7161964,360849848,21378442976,1456505344592,
%T A372278 112197636802224,9643110922761648,914870017865191936,
%U A372278 94969006015521439232,10707303771557931935744,1302965738334245437242368,170216425515761065556430336
%N A372278 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(5/2)) ).
%H A372278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A372278 E.g.f.: A(x) = exp( x - 2/5 * LambertW(-5*x/2 * exp(5*x/2)) ).
%F A372278 If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).
%F A372278 G.f.: Sum_{k>=0} (5*k/2+1)^(k-1) * x^k/(1 - (5*k/2+1)*x)^(k+1).
%F A372278 a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (2^(n-1) * LambertW(exp(-1))^(n + 2/5) * exp(n)). - _Vaclav Kotesovec_, May 06 2024
%o A372278 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-2/5*lambertw(-5*x/2*exp(5*x/2)))))
%o A372278 (PARI) a(n, r=1, t=0, u=5/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));
%o A372278 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (5*k/2+1)^(k-1)*x^k/(1-(5*k/2+1)*x)^(k+1)))
%Y A372278 Cf. A349562, A362694, A362734, A372236, A372235.
%Y A372278 Cf. A349716, A372279.
%K A372278 nonn
%O A372278 0,2
%A A372278 _Seiichi Manyama_, Apr 25 2024