A363182 Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^(n+1).

1, 2, 6, 20, 68, 234, 824, 2956, 10750, 39540, 146864, 550096, 2075432, 7880046, 30086704, 115445028, 444941028, 1721720032, 6686357238, 26051961396, 101810056296, 398962013908, 1567354966200, 6171824148252, 24355381522328, 96304034538898, 381506619687824

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

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

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

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 68*x^4 + 234*x^5 + 824*x^6 + 2956*x^7 + 10750*x^8 + 39540*x^9 + 146864*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 * (2*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A363142, A363183, A363184, A363185.

Cf. A359670.

Sequence in context: A006012 A127152 A279557 * A150120 A360219 A360294

Adjacent sequences: A363179 A363180 A363181 * A363183 A363184 A363185

Keyword

nonn

Author

Paul D. Hanna, May 20 2023

Status

approved