A357404 Coefficients in the power series A(x) such that: 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

1, 4, 32, 324, 3632, 43640, 549472, 7154952, 95563392, 1301943972, 18022506736, 252768034908, 3584103003152, 51294399688504, 739984677348512, 10749373940462452, 157101410692820448, 2308378616597302488, 34080671255517914992, 505321131709023383016, 7521442675843527317728

Offset: 0

Author

Paul D. Hanna, Sep 26 2022

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).

a(n) = Sum_{k=0..n} A357400(n,k) * 4^k, for n >= 0.

Examples

			G.f.: A(x) = 1 + 4*x + 32*x^2 + 324*x^3 + 3632*x^4 + 43640*x^5 + 549472*x^6 + 7154952*x^7 + 95563392*x^8 + 1301943972*x^9 + 18022506736*x^10 + ...
such that
4 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-4*A(x)^3 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...
		

Links

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

Crossrefs

Cf. A357400, A356783, A357402, A357403, A357405.

Programs

PARI

{a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(4 - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); 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 relations.

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

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

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

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

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

(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.