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%I #5 Jun 11 2021 21:15:21
%S 1,1,2,3,8,17,42,107,272,719,1914,5163,14088,38733,107370,299511,
%T 840372,2370020,6714316,19100096,54534696,156230943,448942998,
%U 1293692305,3737568960,10823759093,31413810702,91358248179,266193726712,776989772307,2271695757714,6652074198889
%N G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} (-1)^(k+1) * A(x^k) / k).
%F G.f.: x + x^2 * Product_{n>=1} (1 + x^n)^(2*a(n)).
%F a(n+2) = (2/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k+2).
%t nmax = 32; A[_] = 0; Do[A[x_] = x + x^2 Exp[2 Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t a[1] = a[2] = 1; a[n_] := a[n] = (2/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 32}]
%Y Cf. A005753, A007560, A345244, A345245.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Jun 11 2021