A357797 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.

1, 1, 5, 18, 85, 374, 1659, 7774, 36876, 177494, 867424, 4285653, 21373782, 107475746, 544244911, 2773091748, 14207171278, 73140904609, 378184133959, 1963127909395, 10226682384980, 53446907352828, 280150058149086, 1472424136948438, 7758105323877698, 40970959715619200, 216830651728330127

Offset: 0

Author

Paul D. Hanna, Dec 22 2022

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 85*x^4 + 374*x^5 + 1659*x^6 + 7774*x^7 + 36876*x^8 + 177494*x^9 + 867424*x^10 + 4285653*x^11 + 21373782*x^12 + ...
		

Links

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

Crossrefs

Cf. A359720, A359721, A359723, A357798.

Programs

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 * (2 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

PARI

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

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 * (2 + x^n)^n * A(x)^n.

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

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