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A363312 Expansion of g.f. A(x) satisfying 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 3.

%I A363312 #13 Jun 09 2023 21:25:20
%S A363312 3,8,68,656,6924,77816,912504,11043616,136909712,1729812880,
%T A363312 22193496988,288368706416,3786876943856,50180784019384,
%U A363312 670150485880336,9010466250798080,121871951481594296,1657086342551799752,22637216782139196588,310547100988853539728
%N A363312 Expansion of g.f. A(x) satisfying 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 3.
%C A363312 a(n) == 0 (mod 2^2) for n > 0.
%H A363312 Paul D. Hanna, <a href="/A363312/b363312.txt">Table of n, a(n) for n = 0..200</a>
%F A363312 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A363312 (1) 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
%F A363312 (2) 1/2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
%F A363312 (3) A(x)/2 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
%F A363312 (4) A(x)/2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).
%e A363312 G.f.: A(x) = 3 + 8*x + 68*x^2 + 656*x^3 + 6924*x^4 + 77816*x^5 + 912504*x^6 + 11043616*x^7 + 136909712*x^8 + 1729812880*x^9 + ...
%o A363312 (PARI) {a(n) = my(A=[3]); for(i=1,n, A = concat(A,0);
%o A363312 A[#A] = polcoeff(-2  + 2^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}
%o A363312 for(n=0,30,print1(a(n),", "))
%Y A363312 Cf. A357227, A363141, A363313, A363314, A363315.
%K A363312 nonn
%O A363312 0,1
%A A363312 _Paul D. Hanna_, May 28 2023