%I #10 Jul 06 2021 08:07:02
%S 0,1,1,0,1,1,1,2,3,3,6,8,11,18,26,37,60,87,132,206,310,475,742,1130,
%T 1759,2737,4236,6618,10348,16139,25350,39767,62456,98401,155047,
%U 244570,386639,611298,967874,1534297,2433584,3864154,6141560,9766908,15547187,24766037,39476846
%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} d * a(d) ) * a(n-k).
%F a(n) ~ c * d^n / n^(3/2), where d = 1.646504994482771446591056040381099740295861136174688956979834656... and c = 0.8402317368556115946120005582458627329843217960728964299829... - _Vaclav Kotesovec_, Jul 06 2021
%p a:= proc(n) option remember; `if`(n<4, signum(n-1), add(a(n-k)*
%p add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
%p end:
%p seq(a(n), n=1..47); # _Alois P. Heinz_, Jul 01 2021
%t nmax = 47; A[_] = 0; Do[A[x_] = x^2 + x^3 Exp[Sum[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[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 47}]
%Y Cf. A007562, A218020, A346020.
%K nonn
%O 1,8
%A _Ilya Gutkovskiy_, Jul 01 2021