A366229 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.

1, 1, 2, 4, 10, 23, 55, 138, 349, 904, 2377, 6323, 16993, 46036, 125625, 344973, 952565, 2643257, 7366942, 20613366, 57884187, 163071852, 460769168, 1305466309, 3707928596, 10555941648, 30115379589, 86087330322, 246541672062, 707274898726, 2032285666846

Offset: 0

Author

Paul D. Hanna, Oct 29 2023

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(3*n+2))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 23*x^5 + 55*x^6 + 138*x^7 + 349*x^8 + 904*x^9 + 2377*x^10 + 6323*x^11 + 16993*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A358937, A355866.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n * (x^(3*n+1) - Ser(A))^n ), #A) ); 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 formulas.

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

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

(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(3*n+2) - A(x))^n (trivial).

(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n-1)) / (1 - x^(3*n-1)*A(x))^n (trivial).