A357164 Coefficients in the power series A(x) such that: A(x)^4 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

1, 1, 6, 54, 540, 5925, 68753, 830267, 10324947, 131329213, 1700614790, 22344117822, 297132512955, 3991542148276, 54086668396101, 738390401404546, 10146440406910223, 140227571720595241, 1947883865390758591, 27181029295364007844, 380838895427784827916

Offset

0,3

Comments

Compare to A357154.

Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (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)^4 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

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

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

(4) -A(x)^7 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.

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

Example

G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 540*x^4 + 5925*x^5 + 68753*x^6 + 830267*x^7 + 10324947*x^8 + 131329213*x^9 + 1700614790*x^10 + ...
such that
A(x)^4 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
also
-A(x)^7 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(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)^4 - sum(n=-#A\3-2, #A\3+2, x^(3*n+2) * (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. A357154, A357160, A357161, A357162, A357163, A357165.

Sequence in context: A092472 A228413 A098658 * A357225 A109576 A365843

Adjacent sequences: A357161 A357162 A357163 * A357165 A357166 A357167

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2022

Status

approved