A363313 Expansion of g.f. A(x) satisfying 1/3 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 4.

4, 18, 216, 3006, 46062, 752058, 12824370, 225765756, 4072115322, 74865020256, 1397774141280, 26431211243142, 505157673609054, 9742590254518956, 189370217827381284, 3705934209907310622, 72957899444047650828, 1443901345003970392266, 28710711213830156663136

Offset: 0

Author

Paul D. Hanna, May 28 2023

Keywords

• nonn

Comments

a(n) == 0 (mod 3^2) for n > 0.

Examples

			G.f.: A(x) = 4 + 18*x + 216*x^2 + 3006*x^3 + 46062*x^4 + 752058*x^5 + 12824370*x^6 + 225765756*x^7 + 4072115322*x^8 + 74865020256*x^9 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A357227, A363141, A363312, A363314, A363315.

Programs

PARI

{a(n) = my(A=[4]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-3  + 3^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

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

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

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

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

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