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%I #12 Sep 17 2023 21:28:30
%S 1,1,3,7,20,49,148,391,1167,3309,9974,29083,88851,265815,818676,
%T 2497633,7753214,23968298,75015661,234347623,738220718,2325779822,
%U 7369929088,23374281968,74451070741,237439581606,759696374373,2434110886204,7818771362163,25150994589336,81070450302439
%N G.f.: A(x) = x*exp( Sum_{n>=1} Sum_{d|n} A(d*x^n) / n ).
%e G.f.: A(x) = x + x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 49*x^6 + 148*x^7 + 391*x^8 + ...
%e where
%e log(A(x)/x) = A(x) + (A(x^2) + A(2*x^2))/2 + (A(x^3) + A(3*x^3))/3 + (A(x^4) + A(2*x^4) + A(4*x^4))/4 + (A(x^5) + A(5*x^5))/5 + (A(x^6) + A(2*x^6) + A(3*x^6) + A(6*x^6))/6 + (A(x^7) + A(7*x^7))/7 + (A(x^8) + A(2*x^8) + A(4*x^8) + A(8*x^8))/8 + (A(x^9) + A(3*x^9) + A(9*x^9))/9 + ...
%t a[n_] := Module[{A=x}, For[i=1, i <= n, i++, A=x*Exp[Sum[DivisorSum[k, (A /. x :> #*x^k + x*O[x]^n)&]/k, {k, 1, n}]]]; SeriesCoefficient[A, n]]; Table[Print[an = a[n]]; an, {n, 1, 35}] (* _Jean-François Alcover_, Dec 10 2015, adapted from PARI *)
%o (PARI) {a(n)=local(A=x); for(i=1, n, A=x*exp(sum(k=1, n, sumdiv(k,d,subst(A, x, d*x^k +x*O(x^n)))/k))); polcoeff(A, n)}
%o for(n=1, 35, print1(a(n), ", "))
%Y Cf. A229807.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Oct 16 2013
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