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%I #10 Dec 10 2023 17:51:40
%S 1,1,3,7,22,68,230,790,2809,10151,37346,139116,524160,1993184,7641077,
%T 29497043,114568292,447397474,1755543081,6918285937,27369693199,
%U 108659122477,432762724820,1728625913332,6923326780630,27797059897118,111859099064653,451086075723925,1822628515630354,7377839562543650
%N G.f.: A(x) = exp( Sum_{n>=1} (x^n/n) * (1 + x^n*A(x)^n) / (1 - x^n*A(x)^n)).
%e G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 22*x^4 + 68*x^5 + 230*x^6 + 790*x^7 + 2809*x^8 + 10151*x^9 + 37346*x^10 + 139116*x^11 + 524160*x^12 + ...
%e satisfies
%e log(A(x)) = x*(1+x*A(x))/(1-x*A(x)) + x^2/2*(1+x^2*A(x)^2)/(1-x^2*A(x)^2) + x^3/3*(1+x^3*A(x)^3)/(1-x^3*A(x)^3) + x^4/4*(1+x^4*A(x)^4)/(1-x^4*A(x)^4) + x^5/5*(1+x^5*A(x)^5)/(1-x^5*A(x)^5) + x^6/6*(1+x^6*A(x)^6)/(1-x^6*A(x)^6) + ...
%e Explicitly,
%e log(A(x)) = x + 5*x^2/2 + 13*x^3/3 + 53*x^4/4 + 191*x^5/5 + 761*x^6/6 + 2969*x^7/7 + 11893*x^8/8 + 47677*x^9/9 + 192935*x^10/10 + 783575*x^11/11 + 3197657*x^12/12 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=Vec( exp( sum(m=1, #A+1, x^m/m * (1 + x^m*Ser(A)^m)/(1 - x^m*Ser(A)^m) ) )) ); A[n+1]}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A210043.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 16 2019
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