%I #12 Jun 09 2023 21:26:14

%S 4,18,216,3006,46062,752058,12824370,225765756,4072115322,74865020256,

%T 1397774141280,26431211243142,505157673609054,9742590254518956,

%U 189370217827381284,3705934209907310622,72957899444047650828,1443901345003970392266,28710711213830156663136

%N 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.

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

%H Paul D. Hanna, <a href="/A363313/b363313.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

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

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

%e 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 + ...

%o (PARI) {a(n) = my(A=[4]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff(-3 + 3^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}

%o for(n=0,30,print1(a(n),", "))

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

%K nonn

%O 0,1

%A _Paul D. Hanna_, May 28 2023