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 A351156 Expansion of e.g.f. (1 - x^3/6)^(-x). 2

%I

%S 1,0,0,0,4,0,0,70,560,0,5600,92400,369600,1201200,30830800,252252000,

%T 1210809600,19059040000,240143904000,1738184448000,22451549120000,

%U 342205063200000,3417705170880000,43866126368064000,732641268463104000,9234973972224000000

%N Expansion of e.g.f. (1 - x^3/6)^(-x).

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+2)/3)} (3*k-2)/((k-1) * 6^(k-1)) * a(n-3*k+2)/(n-3*k+2)!.

%F a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(k,n-3*k)|/(6^k*k!).

%F a(n) ~ sqrt(2*Pi) * n^(n - 1/2 + 6^(1/3)) / (Gamma(6^(1/3)) * 3^(6^(1/3)) * exp(n) * 6^(n/3)). - _Vaclav Kotesovec_, May 04 2022

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3/6)^(-x)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^3/6))))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, (i+2)\3, (3*j-2)/((j-1)*6^(j-1))*v[i-3*j+3]/(i-3*j+2)!)); v;

%o (PARI) a(n) = n!*sum(k=0, n\3, abs(stirling(k, n-3*k, 1))/(6^k*k!));

%Y Cf. A351155, A353227.

%K nonn

%O 0,5

%A _Seiichi Manyama_, May 02 2022

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Last modified March 29 21:39 EDT 2023. Contains 361599 sequences. (Running on oeis4.)