A357202 Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

1, 1, 2, 9, 35, 182, 921, 5062, 28234, 162330, 947773, 5622641, 33747694, 204676547, 1252083028, 7717376754, 47878314072, 298749048454, 1873637869199, 11804288518884, 74673607921030, 474128308291896, 3020493580980524, 19301224674496592, 123681469340775568

Offset

0,3

Comments

Compare to A357152 and A357162.

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

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

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

(4) -A(x)^5 = 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 + 2*x^2 + 9*x^3 + 35*x^4 + 182*x^5 + 921*x^6 + 5062*x^7 + 28234*x^8 + 162330*x^9 + 947773*x^10 + 5622641*x^11 + 33747694*x^12 + ...
such that
A(x)^2 = ... + 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)^5 = ... + 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)^2 - 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. A357152, A357162, A357200, A357201, A357203, A357204, A357205.

Sequence in context: A150966 A185034 A185234 * A306791 A175231 A175261

Adjacent sequences: A357199 A357200 A357201 * A357203 A357204 A357205

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2022

Status

approved