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A359723 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.

%I A359723 #8 Jan 14 2023 10:08:08
%S A359723 1,1,7,28,151,803,4108,22532,125449,705929,4035955,23332364,136111591,
%T A359723 800561116,4741777880,28258286033,169322163149,1019483819757,
%U A359723 6164900341534,37425357962592,228002416106605,1393503512669230,8541839907812651,52500559705299795,323483846045526418
%N A359723 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.
%C A359723 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%F A359723 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A359723 (1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.
%F A359723 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 3*x^n)^n * A(x)^n).
%F A359723 (3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*3^k, for n >= 0.
%e A359723 G.f.: A(x) = 1 + x + 7*x^2 + 28*x^3 + 151*x^4 + 803*x^5 + 4108*x^6 + 22532*x^7 + 125449*x^8 + 705929*x^9 + 4035955*x^10 + 23332364*x^11 + 136111591*x^12 + ...
%e A359723 SPECIFIC VALUES.
%e A359723 A(x) = 2 at x = 0.150684304746792807618050217238804920801612774142866...
%e A359723 A(1/7) = 1.67848119643298635311797131334138331526984303696733717...
%e A359723 A(1/8) = 1.40389487408504106142147713148599989460789630965507028...
%o A359723 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A359723 A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (3 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
%o A359723 for(n=0, 30, print1(a(n), ", "))
%Y A359723 Cf. A359720, A355357, A357797, A359721, A359724.
%K A359723 nonn
%O A359723 0,3
%A A359723 _Paul D. Hanna_, Jan 11 2023