A358962 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).

1, 2, 8, 30, 146, 748, 4002, 22114, 125220, 722850, 4238148, 25169064, 151084168, 915235106, 5587985801, 34351213384, 212436911849, 1320744403708, 8250065775120, 51752790871466, 325887027304769, 2059216160242430, 13052805881695018, 82976612756731258

Offset: 0

Author

Paul D. Hanna, Dec 07 2022

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 + 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 + ...
where A = A(x) satisfies the doubly infinite sum
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) + ...
also,
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) + ...
		

Links

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

Crossrefs

Cf. A358961, A358963, A358964, A358965.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(3*n+2))^(n-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 satisfies:

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

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

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

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

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