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A346020 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

%I A346020 #12 Jul 06 2021 07:45:38
%S A346020 1,0,1,1,1,2,3,4,7,11,17,28,46,74,124,206,343,577,976,1649,2808,4792,
%T A346020 8200,14073,24228,41782,72246,125164,217262,377784,658072,1148006,
%U A346020 2005743,3509125,6147422,10782375,18934209,33285291,58575080,103181405,181928014,321059155
%N A346020 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).
%H A346020 Vaclav Kotesovec, <a href="/A346020/b346020.txt">Table of n, a(n) for n = 1..3800</a>
%F A346020 G.f.: x + x^3 / Product_{n>=1} (1 - x^n)^a(n).
%F A346020 a(1) = 1, a(2) = 0, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} d * a(d) ) * a(n-k).
%F A346020 a(n) ~ c * d^n / n^(3/2), where d = 1.82975393308934955558864748939303527364978309460948926333116466766295641... and c = 0.8335864368398390652263577663136791087027831725508605623969711758177... - _Vaclav Kotesovec_, Jul 06 2021
%p A346020 a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*
%p A346020        add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
%p A346020     end:
%p A346020 seq(a(n), n=1..42);  # _Alois P. Heinz_, Jul 01 2021
%t A346020 nmax = 42; A[_] = 0; Do[A[x_] = x + x^3 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t A346020 a[1] = 1; a[2] = 0; 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, 42}]
%Y A346020 Cf. A007562, A218020, A346030.
%K A346020 nonn
%O A346020 1,6
%A A346020 _Ilya Gutkovskiy_, Jul 01 2021