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G.f. A(x) satisfies: x = Sum_{n>=1} ((1+x)^n - 1)^n / A(x)^n.
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%I #9 May 04 2018 18:20:44

%S 1,4,15,110,1319,21626,440406,10567338,289567071,8881182506,

%T 300677809182,11123151634732,446124758009229,19278179994562787,

%U 892894885467043254,44130236756271133940,2318466084626196805383,129037338117765390406606,7585054768393048247917776,469612308887467564648834414

%N G.f. A(x) satisfies: x = Sum_{n>=1} ((1+x)^n - 1)^n / A(x)^n.

%H Paul D. Hanna, <a href="/A303654/b303654.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: A(x) = (1+x) / ( Sum_{n>=0} (1+x)^(n^2) / ((1+x)^n + A(x))^(n+1) ).

%e G.f.: A(x) = 1 + 4*x + 15*x^2 + 110*x^3 + 1319*x^4 + 21626*x^5 + 440406*x^6 + 10567338*x^7 + 289567071*x^8 + 8881182506*x^9 + ...

%e such that

%e x = ((1+x) - 1)/A(x) + ((1+x)^2 - 1)^2/A(x)^2 + ((1+x)^3 - 1)^3/A(x)^3 + ((1+x)^4 - 1)^4/A(x)^4 + ((1+x)^5 - 1)^5/A(x)^5 + ((1+x)^6 - 1)^6/A(x)^6 + ...

%e Also,

%e A(x) = (1+x) / ( 1/(1 + A(x))) + (1+x)/((1+x) + A(x))^2 + (1+x)^4/((1+x)^2 + A(x))^3 + (1+x)^9/((1+x)^3 + A(x))^4 + (1+x)^16/((1+x)^4 + A(x))^5 + (1+x)^25/((1+x)^5 + A(x))^6 + (1+x)^36/((1+x)^6 + A(x))^7 + ... ).

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(m=1, #A, ((1+x)^m - 1 +x*O(x^#A))^m / Ser(A)^m ) )[#A] ); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 04 2018