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A346031 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 A346031 #9 Jul 01 2021 19:35:15
%S A346031 1,0,1,1,0,1,2,1,1,4,4,3,9,14,12,22,43,49,66,130,186,234,406,663,884,
%T A346031 1362,2303,3347,4884,8049,12478,18240,28853,46075,69163,106470,170305,
%U A346031 262853,401773,635780,998609,1536093,2405345,3801601,5910267,9212253,14548179,22818301
%N A346031 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 + ...).
%F A346031 G.f.: x + x^3 * Product_{n>=1} (1 + x^n)^a(n).
%F A346031 a(1) = 1, a(2) = 0, 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 A346031 a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*add(
%p A346031      (-1)^(k/d+1)*d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
%p A346031     end:
%p A346031 seq(a(n), n=1..48);  # _Alois P. Heinz_, Jul 01 2021
%t A346031 nmax = 48; A[_] = 0; Do[A[x_] = x + 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 A346031 a[1] = 1; a[2] = 0; 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, 48}]
%Y A346031 Cf. A007560, A316075, A346032.
%K A346031 nonn
%O A346031 1,7
%A A346031 _Ilya Gutkovskiy_, Jul 01 2021