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Expansion of e.g.f. 1/(1 + x/8 * log(1 - 4 * x)).
3

%I #10 May 24 2022 08:12:09

%S 1,0,1,6,70,1080,21162,501060,13904152,442241856,15855648120,

%T 632501646480,27781645311216,1332152096109120,69237728070951888,

%U 3876953348374273440,232666700169003442560,14897335773169370787840,1013656610215024983681408

%N Expansion of e.g.f. 1/(1 + x/8 * log(1 - 4 * x)).

%F a(0) = 1; a(n) = (n!/2) * Sum_{k=2..n} 4^(k-2)/(k-1) * a(n-k)/(n-k)!.

%F a(n) = n! * Sum_{k=0..floor(n/2)} 4^(n-2*k) * k! * |Stirling1(n-k,k)|/(2^k * (n-k)!).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x/8*log(1-4*x))))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 4^(j-2)/(j-1)*v[i-j+1]/(i-j)!)/2); v;

%o (PARI) a(n) = n!*sum(k=0, n\2, 4^(n-2*k)*k!*abs(stirling(n-k, k, 1))/(2^k*(n-k)!));

%Y Cf. A354326, A354327.

%K nonn

%O 0,4

%A _Seiichi Manyama_, May 24 2022