|
%I #4 Aug 13 2020 08:58:39
%S 1,0,2,9,112,1875,43416,1310946,49778688,2313362673,128894500000,
%T 8469572721533,647341071298560,56871349337125648,5684260661585401728,
%U 640631299771142578125,80788871646072851660800,11323828537291632967145015,1753760620207362607774290432
%N a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (n*k - 1)^n / (n^k * k!).
%F a(n) = [x^n] (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - n*j*x/(1 + x)).
%F a(n) = n! * [x^n] exp((exp(n*x) - 1) / n - x), for n > 0.
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * n^k * BellPolynomial_k(1/n), for n > 0.
%t Table[SeriesCoefficient[1/(1 + x) Sum[(x/(1 + x))^k/Product[(1 - n j x/(1 + x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
%t Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n - x], {x, 0, n}], {n, 1, 18}]]
%t Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n,k] n^k BellB[k, 1/n], {k, 0, n}], {n, 1, 18}]]
%Y Cf. A000296, A301419, A330605, A334162, A337038, A337039, A337040, A337041, A337042.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 12 2020
|
