A372236 - OEIS

%I #23 Apr 24 2024 07:01:49

%S 1,2,6,26,152,1132,10300,111064,1387104,19713104,314350064,5560881328,

%T 108110428288,2291750937088,52618613073408,1301031907140608,

%U 34470409922547200,974354631630161152,29270099764874881792,931275451933870415104,31285710787985504633856

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

%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( x - 2*LambertW(-x/2 * exp(x/2)) ).

%F 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 G.f.: Sum_{k>=0} (k/2+1)^(k-1) * x^k/(1 - (k/2+1)*x)^(k+1).

%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^(n+2)). - _Vaclav Kotesovec_, Apr 24 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-2*lambertw(-x/2*exp(x/2)))))

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (k/2+1)^(k-1)*x^k/(1-(k/2+1)*x)^(k+1)))

%Y Cf. A007889, A371524.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 23 2024