A357205 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

1, 1, 5, 45, 453, 5072, 59964, 738449, 9365617, 121511799, 1605113475, 21514501261, 291880434822, 4000334186684, 55304105835751, 770323876417969, 10800108248187952, 152293211204657100, 2158477865404685913, 30732066480408276249, 439351185869943970405

Offset

0,3

Comments

Compare to A357155 and A357165.

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

Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Links

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

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.

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

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

(3) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^n*A(x))^n.

(4) -A(x)^8 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(n+1))^(n+1) / A(x)^n.

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

Example

G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 453*x^4 + 5072*x^5 + 59964*x^6 + 738449*x^7 + 9365617*x^8 + 121511799*x^9 + 1605113475*x^10 + ...
such that
A(x)^5 = ... + x^(-2)*(1 - 1/x)^(-1)/A(x)^2 + x^(-1)/A(x) + (1 - x) + x*(1 - x^2)*A(x) + x^2*(1 - x^3)^3*A(x)^2 + x^3*(1 - x^4)^4*A(x)^3 + ... + x^n*(1 - x^(n+1))^(n+1)*A(x)^n + ...
also
-A(x)^8 = ... + x^(-2)*(A(x) - 1/x)^(-1)*A(x)^2 + x^(-1)*A(x) + (A(x) - x) + x*(A(x) - x^2)^2/A(x) + x^2*(A(x) - x^3)^3/A(x)^2 + x^3*(A(x) - x^4)^4/A(x)^3 + ... + x^n*(A(x) - x^(n+1))^(n+1)/A(x)^n + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(Ser(A)^5 - sum(n=-#A-2, #A+2, x^(n) * (1 - x^(n+1) +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A357155, A357165, A357200, A357201, A357202, A357203, A357204.

Sequence in context: A054318 A093140 A137233 * A001449 A371756 A209442

Adjacent sequences: A357202 A357203 A357204 * A357206 A357207 A357208

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2022

Status

approved