A359673 a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n*A(x)^n)^n.

1, 2, 5, 13, 30, 74, 202, 616, 2126, 7828, 29366, 110398, 414214, 1556848, 5892713, 22524354, 86954484, 338421674, 1324660464, 5204326208, 20498580511, 80907096678, 320002290542, 1268500509496, 5040195484362, 20073242195580, 80120884387322, 320442284717582, 1283939790460139

Offset: 0

Author

Paul D. Hanna, Jan 10 2023

Keywords

• nonn

Comments

Given g.f. A(x), x*A(x) equals a series reversion of x*G(-x) where G(x) is the g.f. of A355868.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 13*x^3 + 30*x^4 + 74*x^5 + 202*x^6 + 616*x^7 + 2126*x^8 + 7828*x^9 + 29366*x^10 + 110398*x^11 + 414214*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.2170550872218893465015254812376904599677836767029937...
A(1/5) = 1.8185729641608353079390837085677719656772552871159724...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A355868.

Programs

PARI

{a(n) = my(A=[1]);
for(i=1,n, A = concat(A,0); A[#A] = polcoeff(-1 + sum(m=-#A,#A, (2*x + (-x*Ser(A))^m)^m ), #A)/2);A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

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

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

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

(4) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 + 2*(-x)^(n+1) * A(x)^n)^(n+1).