A363183 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^(n+1).

1, 3, 11, 45, 193, 846, 3779, 17169, 79115, 368820, 1736169, 8241039, 39400672, 189567594, 917146729, 4459208292, 21776797603, 106771412718, 525382657858, 2593665077634, 12842387591191, 63762186132387, 317373771999035, 1583380006374078, 7916456438276103

Offset

0,2

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.

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

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

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

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

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

a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 3^(n-2*k) for n >= 0.

Example

G.f.: A(x) = 1 + 3*x + 11*x^2 + 45*x^3 + 193*x^4 + 846*x^5 + 3779*x^6 + 17169*x^7 + 79115*x^8 + 368820*x^9 + 1736169*x^10 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (3*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/3); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A363142, A363182, A363184, A363185.

Cf. A359670.

Sequence in context: A151113 A151114 A083878 * A151115 A151116 A151117

Adjacent sequences: A363180 A363181 A363182 * A363184 A363185 A363186

Keyword

nonn

Author

Paul D. Hanna, May 20 2023

Status

approved