%I #20 Nov 28 2021 02:49:19
%S 1,1,6,91,2156,69926,2884576,144555356,8529135216,579220982056,
%T 44503081624976,3816776859516776,361462121953291456,
%U 37464997600663289216,4218485281787859411456,512762346462142021355776,66919363061333997572830976,9332997074366800051673277056
%N E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^5)/2 ).
%H Seiichi Manyama, <a href="/A349716/b349716.txt">Table of n, a(n) for n = 0..330</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = (1/2^n) * Sum_{k=0..n} (5*k+1)^(n-1) * binomial(n,k).
%F E.g.f.: ( -LambertW( -5*x/2 * exp(5*x/2) )/(5*x/2) )^(1/5).
%F G.f.: 2 * Sum_{k>=0} (5*k+1)^(k-1) * x^k/(2 - (5*k+1)*x)^(k+1).
%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/5) * 2^n * exp(n)). - _Vaclav Kotesovec_, Nov 26 2021
%t a[n_] := (1/2^n) * Sum[(5*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o (PARI) a(n) = sum(k=0, n, (5*k+1)^(n-1)*binomial(n, k))/2^n;
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-5*x/2*exp(5*x/2))/(5*x/2))^(1/5)))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (5*k+1)^(k-1)*x^k/(2-(5*k+1)*x)^(k+1)))
%Y Cf. A007889, A202617, A349714, A349715, A349719, A349720, A349721.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 26 2021