A359724 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.

1, 1, 9, 40, 235, 1456, 8323, 51510, 324674, 2061746, 13308492, 86876405, 572169044, 3799139674, 25403610485, 170901457100, 1155976005944, 7856772779823, 53630378512469, 367507023955203, 2527254094342404, 17435029150904202, 120633291776867632, 836907189915348056

Offset

0,3

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Links

Table of n, a(n) for n=0..23.

Formula

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

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

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

(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*4^k, for n >= 0.

Example

G.f.: A(x) = 1 + x + 9*x^2 + 40*x^3 + 235*x^4 + 1456*x^5 + 8323*x^6 + 51510*x^7 + 324674*x^8 + 2061746*x^9 + 13308492*x^10 + 86876405*x^11 + 572169044*x^12 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (4 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A359720, A355357, A357797, A359721, A359723.

Sequence in context: A026629 A243718 A086349 * A026651 A129799 A357552

Adjacent sequences: A359721 A359722 A359723 * A359725 A359726 A359727

Keyword

nonn

Author

Paul D. Hanna, Jan 11 2023

Status

approved