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%I #19 Jan 25 2015 08:19:30
%S 1,1,2,9,45,260,1631,10901,76489,558396,4215058,32758362,261329689,
%T 2135425660,17847456953,152421972422,1329377420317,11837472015705,
%U 107612097309239,998877348931934,9469405684352153,91713801856207441,907821847607245454,9186799075352185868,95069881782485132500
%N G.f. satisfies: A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n / (1+x)^(n+1).
%H Vaclav Kotesovec, <a href="/A251178/b251178.txt">Table of n, a(n) for n = 0..128</a>
%F G.f. satisfies:
%F (1) A(x) = Sum_{n>=0} A(x)^(n^2) * x^n / (1+x - x*A(x)^n)^(n+1).
%F (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (-1)^(n-k) * (A(x)^k + 1)^k.
%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 45*x^4 + 260*x^5 + 1631*x^6 +...
%e where we have the identities:
%e (0) A(x) = 1/(1+x) + (A(x)+1)*x/(1+x)^2 + (A(x)^2+1)^2*x^2/(1+x)^3 + (A(x)^3+1)^3*x^3/(1+x)^4 + (A(x)^4+1)^4*x^4/(1+x)^5 + (A(x)^5+1)^5*x^5/(1+x)^6 +...
%e (1) A(x) = 1 + A(x)*x/(1+x - x*A(x))^2 + A(x)^4*x^2/(1+x - x*A(x)^2)^3 + A(x)^9*x^3/(1+x - x*A(x)^3)^4 + A(x)^16*x^4/(1+x - x*A(x)^4)^5 + A(x)^25*x^5/(1+x - x*A(x)^5)^6 + A(x)^36*x^6/(1+x - x*A(x)^6)^7 +...
%e (2) A(x) = 1 - x*(1 - (A(x)+1)) + x^2*(1 - 2*(A(x)+1) + (A(x)^2+1)^2) - x^3*(1 - 3*(A(x)+1) + 3*(A(x)^2+1)^2 - (A(x)^3+1)^3) + x^4*(1 - 4*(A(x)+1) + 6*(A(x)^2+1)^2 - 4*(A(x)^3+1)^3 + (A(x)^4+1)^4) - x^5*(1 - 5*(A(x)+1) + 10*(A(x)^2+1)^2 - 10*(A(x)^3+1)^3 + 5*(A(x)^4+1)^4 - (A(x)^5+1)^5) +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (A^m + 1)^m * x^m / (1+x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, A^(m^2) * x^m / (1+x - x*A^m +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m * sum(k=0, m, binomial(m, k) * (-1)^(m-k) * (A^k + 1)^k +x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A251177, A251179, A251180, A251181, A244610.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 19 2015
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