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%I #8 Jul 01 2021 19:35:07
%S 0,1,1,0,1,1,0,2,1,1,4,2,4,9,4,14,20,15,43,48,55,127,127,199,363,379,
%T 684,1048,1229,2263,3100,4163,7288,9558,14231,23222,30673,48404,74113,
%U 101631,163048,239282,343196,545318,785139,1169148,1818866,2619072,3991888,6079434
%N G.f. A(x) satisfies: A(x) = x^2 + x^3 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).
%F G.f.: x^2 + x^3 * Product_{n>=1} (1 + x^n)^a(n).
%F a(1) = 0, a(2) = 1, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k).
%p a:= proc(n) option remember; `if`(n<4, signum(n-1), add(a(n-k)*add(
%p (-1)^(k/d+1)*d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
%p end:
%p seq(a(n), n=1..50); # _Alois P. Heinz_, Jul 01 2021
%t nmax = 50; A[_] = 0; Do[A[x_] = x^2 + x^3 Exp[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] = 0; a[2] = 1; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 50}]
%Y Cf. A007560, A316075, A346031.
%K nonn
%O 1,8
%A _Ilya Gutkovskiy_, Jul 01 2021
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