%I #21 Dec 09 2022 14:33:22

%S 1,2,8,30,146,748,4002,22114,125220,722850,4238148,25169064,151084168,

%T 915235106,5587985801,34351213384,212436911849,1320744403708,

%U 8250065775120,51752790871466,325887027304769,2059216160242430,13052805881695018,82976612756731258

%N a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(3*n+2))^(n-1).

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

%H Paul D. Hanna, <a href="/A358962/b358962.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

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

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

%F (3) A(x) = Sum_{n=-oo..+oo} x^(4*n+2)* (A(x) - x^(3*n+2))^(n-1).

%F (4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n*(n-1)) / (1 - x^(3*n-2)*A(x))^(n+1).

%F (5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n*(n-1)) / (1 - x^(3*n-2)*A(x))^n.

%e G.f.: A(x) = 1 + 2*x + 8*x^2 + 30*x^3 + 146*x^4 + 748*x^5 + 4002*x^6 + 22114*x^7 + 125220*x^8 + 722850*x^9 + 4238148*x^10 + ...

%e where A = A(x) satisfies the doubly infinite sum

%e 1 = ... + x^(-2)*(A - x^(-4))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^2)^(-1) + x*(A - x^5)^0 + x^2*(A - x^8) + x^3*(A - x^11)^2 + x^4*(A - x^14)^3 + ... + x^n * (A - x^(3*n+2))^(n-1) + ...

%e also,

%e A(x) = ... + x^36/(1 - x^(-11)*A)^(-2) - x^18/(1 - x^(-8)*A)^(-1) + x^6 - 1/(1 - x^(-2)*A) + 1/(1 - x*A)^2 - x^6/(1 - x^4*A)^3 + x^18/(1 - x^7*A)^4 - x^36/(1 - x^10*A)^5 + ... + (-1)^(n+1)*x^(3*n*(n-1))/(1 - x^(3*n-2)*A)^(n+1) + ...

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

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

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

%Y Cf. A358961, A358963, A358964, A358965.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 07 2022