%I #6 May 13 2023 13:27:55

%S 1,1,1,7,28,89,421,1898,7912,36412,169960,779139,3668210,17486938,

%T 83333003,400956919,1943928504,9455346485,46225027071,227066384875,

%U 1119123274755,5534782142253,27463607765186,136652474592260,681728348606011,3409395265172439,17088672210734316

%N Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(n-1).

%H Paul D. Hanna, <a href="/A361771/b361771.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

%F (1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(n-1).

%F (2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*A(x)*(-x)^n)^(n+1).

%e G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 28*x^4 + 89*x^5 + 421*x^6 + 1898*x^7 + 7912*x^8 + 36412*x^9 + 169960*x^10 + 779139*x^11 + 3668210*x^12 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);

%o A[#A] = polcoeff( sum(m=-#A, #A, x^m * (2*Ser(A) - (-x)^m)^(m-1) ), #A-1)/2); A[n+1]}

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

%Y Cf. A361772, A361773, A361774.

%Y Cf. A355865, A357227, A359712, A357232.

%K nonn

%O 0,4

%A _Paul D. Hanna_, May 13 2023