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%I #11 Jan 18 2025 09:06:23
%S 1,1,-2,16,-206,3682,-84236,2348704,-77241380,2926735516,
%T -125540336024,6013069027648,-318093606114536,18418565715581656,
%U -1158626159228481488,78679416565851286144,-5736477278907382585328,446936684375920051751440,-37056888825921886749507872
%N Expansion of e.g.f. exp( ((1+5*x)^(2/5) - 1)/2 ).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F a(n) = Sum_{k=0..n} 5^(n-k) * Stirling1(n,k) * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * Stirling1(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
%F a(n) = (1/exp(1/2)) * 5^n * n! * Sum_{k>=0} binomial(2*k/5,n)/(2^k * k!).
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(((1+5*x)^(2/5)-1)/2)))
%Y Cf. A000085, A002119, A380261.
%K sign,new
%O 0,3
%A _Seiichi Manyama_, Jan 18 2025