%I #13 Dec 03 2022 12:05:10
%S 1,1,1,5,37,367,4463,63797,1043961,19208815,392278493,8802891869,
%T 215335062049,5704017709585,162695460126735,4972552233126827,
%U 162156046298476305,5620675413587870585,206382551428754263839,8003189847508668434429
%N a(n) = coefficient of x^n, n >= 0, in A(x) such that: 0 = Sum_{n>=1} ((1+x)^n - A(x))^n / (1+x)^(n^2).
%C All terms appear to be odd.
%H Paul D. Hanna, <a href="/A357397/b357397.txt">Table of n, a(n) for n = 0..300</a>
%e G.f.: A(X) = 1 + x + x^2 + 5*x^3 + 37*x^4 + 367*x^5 + 4463*x^6 + 63797*x^7 + 1043961*x^8 + 19208815*x^9 + 392278493*x^10 + ...
%e where
%e 0 = ((1+x) - A(x))/(1+x) + ((1+x)^2 - A(x))^2/(1+x)^4 + ((1+x)^3 - A(x))^3/(1+x)^9 + ((1+x)^4 - A(x))^4/(1+x)^16 + ((1+x)^5 - A(x))^5/(1+x)^25 + ... + ((1+x)^n - A(x))^n/(1+x)^(n^2) + ...
%e equivalently,
%e 0 = (1 - A(x)/(1+x)) + (1 - A(x)/(1+x)^2)^2 + (1 - A(x)/(1+x)^3)^3 + (1 - A(x)/(1+x)^4)^4 + (1 - A(x)/(1+x)^5)^5 + ... + (1 - A(x)/(1+x)^n)^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=1,#A-1, ((1+x)^m - Ser(A))^m/(1+x +x*O(x^#A) )^(m^2) ),#A-1) ); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A357398.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Oct 20 2022