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.

1, 1, 7, 28, 151, 803, 4108, 22532, 125449, 705929, 4035955, 23332364, 136111591, 800561116, 4741777880, 28258286033, 169322163149, 1019483819757, 6164900341534, 37425357962592, 228002416106605, 1393503512669230, 8541839907812651, 52500559705299795, 323483846045526418

Offset: 0

Author

Paul D. Hanna, Jan 11 2023

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			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 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.150684304746792807618050217238804920801612774142866...
A(1/7) = 1.67848119643298635311797131334138331526984303696733717...
A(1/8) = 1.40389487408504106142147713148599989460789630965507028...
		

Crossrefs

Cf. A359720, A355357, A357797, A359721, A359724.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.

(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.

(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 3*x^n)^n * A(x)^n).

(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*3^k, for n >= 0.