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%I #3 Jan 30 2019 17:52:06
%S 1,3,10,36,155,825,5227,36930,277933,2181186,17716100,148600698,
%T 1286066999,11471959881,105320978028,993555579651,9616293352921,
%U 95370420276468,968267603845781,10056629892262203,106798045980208919,1159174549401667188,12854190552592134081,145574358625296958326,1683040220764208777635
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (3 + x*A(x)^n)^n.
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} x^n * (3 + x*A(x)^n)^n.
%F (2) A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2) / (1 - 3*x*A(x)^n)^(n+1).
%e G.f.: A(x) = 1 + 3*x + 10*x^2 + 36*x^3 + 155*x^4 + 825*x^5 + 5227*x^6 + 36930*x^7 + 277933*x^8 + 2181186*x^9 + 17716100*x^10 + ...
%e such that
%e A(x) = 1 + x*(3 + x*A(x)) + x^2*(3 + x*A(x)^2)^2 + x^3*(3 + x*A(x)^3)^3 + x^4*(3 + x*A(x)^4)^4 + x^5*(3 + x*A(x)^5)^5 + x^6*(3 + x*A(x)^6)^6 + ...
%e Also, the g.f. satisfies the identity:
%e A(x) = 1/(1 - 3*x) + x^2*A(x)/(1 - 3*x*A(x))^2 + x^4*A(x)^4/(1 - 3*x*A(x)^2)^3 + x^6*A(x)^9/(1 - 3*x*A(x)^3)^4 + x^8*A(x)^16/(1 - 3*x*A(x)^4)^5 + x^10*A(x)^25/(1 - 3*x*A(x)^5)^6 + ...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(3 + x*(A+x*O(x^n))^m)^m)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^(2*k)*A^(k^2)/(1 - 3*x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A186998, A203014, A300049, A322725.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 30 2019
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