A357200 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

1, 1, 0, 0, -7, -3, -17, 52, 51, 384, -227, -52, -6311, -2722, -18733, 79229, 67453, 620385, -619315, 85796, -13137380, -595833, -43282243, 205480697, 66895157, 1551910768, -2300631561, 1546386060, -36481481081, 15982958026, -135266506195, 652843485153

Offset

0,5

Comments

Compare to A356783 and A357160.

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

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

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

(4) -A(x)^3 = 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 - 7*x^4 - 3*x^5 - 17*x^6 + 52*x^7 + 51*x^8 + 384*x^9 - 227*x^10 - 52*x^11 - 6311*x^12 - 2722*x^13 - 18733*x^14 + ...
such that
1 = ... + 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)^3 = ... + 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(1 - 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. A356783, A357160, A357201, A357202, A357203, A357204, A357205.

Sequence in context: A200943 A357113 A283289 * A284477 A145399 A356890

Adjacent sequences: A357197 A357198 A357199 * A357201 A357202 A357203

Keyword

sign

Author

Paul D. Hanna, Sep 17 2022

Status

approved