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%I #11 Mar 11 2022 15:24:22
%S 1,1,2,5,10,20,43,93,194,403,842,1755,3656,7643,15976,33281,69164,
%T 143558,297619,616625,1277729,2647861,5485300,11356731,23495794,
%U 48567063,100301668,206994479,426941231,880227976,1814221503,3738368348,7701376466
%N G.f. A(x) satisfies: Product_{n>=1} (1 + x^n*A(x)) = Product_{n>=1} (1 + x^n/(1-x)^n).
%H Paul D. Hanna, <a href="/A352120/b352120.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n and P(x) = Product_{n>=1} (1 + x^n/(1-x)^n) satisfies:
%F (1) P(x) = Product_{n>=1} (1 + x^n*A(x)).
%F (2) P(x) = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^n / (Product_{k=1..n} (1 - x^k)).
%F (3) 1/P(x) = Sum_{n>=0} (-x)^n * A(x)^n / (Product_{k=1..n} (1 - x^k)).
%F (4) log(P(x)) = Sum_{n>=1} x^n * Sum_{d|n} -(-A(x))^(n/d) * d/n.
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 20*x^5 + 43*x^6 + 93*x^7 + 194*x^8 + 403*x^9 + 842*x^10 + 1755*x^11 + 3656*x^12 + ...
%e such that the following products are equal:
%e P(x) = (1 + x*A(x)) * (1 + x^2*A(x)) * (1 + x^3*A(x)) * (1 + x^4*A(x)) * (1 + x^5*A(x)) * (1 + x^6*A(x)) * ...
%e P(x) = (1 + x/(1-x)) * (1 + x^2/(1-x)^2) * (1 + x^3/(1-x)^3) * (1 + x^4/(1-x)^4) * (1 + x^5/(1-x)^5) * ...
%e also, we have the sums
%e P(x) = 1 + x*A(x)/(1-x) + x^3*A(x)^2/((1-x)*(1-x^2)) + x^6*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...
%e 1/P(x) = 1 - x*A(x)/(1-x) + x^2*A(x)^2/((1-x)*(1-x^2)) - x^3*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^4*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+ ...
%e where
%e P(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 28*x^5 + 65*x^6 + 151*x^7 + 350*x^8 + 807*x^9 + 1850*x^10 + ... + A129519(n)*x^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( prod(n=1,#A, (1 + x^n/(1-x +x*O(x^#A))^n)/(1 + x^n*Ser(A)) ),#A) );A[n+1]}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A129519.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 05 2022
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