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A363104 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-x)^n * (4*A(x) + x^(n-1))^(n+1).

%I A363104 #12 May 22 2023 08:57:27
%S A363104 1,6,44,348,2886,24800,218888,1972572,18075100,167900506,1577467760,
%T A363104 14963979584,143124912880,1378756186748,13365212659144,
%U A363104 130274948580864,1276075285222662,12554452588117632,124003727286837484,1229203475053859456,12224294019862383720
%N A363104 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-x)^n * (4*A(x) + x^(n-1))^(n+1).
%C A363104 Conjecture: g.f. A(x) == theta_3(x) (mod 4); a(n) == 2 (mod 4) iff n is a nonzero square and a(n) == 0 (mod 4) iff n is nonsquare.
%H A363104 Paul D. Hanna, <a href="/A363104/b363104.txt">Table of n, a(n) for n = 0..300</a>
%F A363104 G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F A363104 (1) 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(n-1))^(n+1).
%F A363104 (2) 4 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (4*A(x) + x^n)^n.
%F A363104 (3) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^(n-1).
%F A363104 (4) 4*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^(n+1).
%F A363104 (5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(n-1))^n ].
%F A363104 (6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (4*A(x) + x^n)^n ].
%F A363104 (7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^(n+1))^n ].
%F A363104 (8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^n)^(n+1).
%F A363104 (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^n)^n.
%F A363104 (10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^n.
%F A363104 a(n) = Sum_{k=0..n} A359670(n,k) * 4^k for n >= 0.
%e A363104 G.f.: A(x) = 1 + 6*x + 44*x^2 + 348*x^3 + 2886*x^4 + 24800*x^5 + 218888*x^6 + 1972572*x^7 + 18075100*x^8 + 167900506*x^9 + 1577467760*x^10 + ...
%o A363104 (PARI) {a(n) = my(A=1, y=4); for(i=1, n,
%o A363104 A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
%o A363104 polcoeff( A, n, x)}
%o A363104 for(n=0, 25, print1( a(n), ", "))
%o A363104 (PARI) {a(n) = my(A=[1], y=4); for(i=1, n, A = concat(A, 0);
%o A363104 A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
%o A363104 for(n=0, 25, print1( a(n), ", "))
%Y A363104 Cf. A359670, A359711, A359712, A359713, A363105.
%Y A363104 Cf. A363184.
%K A363104 nonn
%O A363104 0,2
%A A363104 _Paul D. Hanna_, May 21 2023