A217699 G.f. A(x) satisfies: 1 - x*A(x) + x^2*A(x)^2 = Sum_{n>=0} (-x)^(n^2).

1, 1, 2, 4, 12, 36, 112, 360, 1185, 3970, 13510, 46564, 162212, 570256, 2020512, 7208015, 25868510, 93331707, 338328434, 1231650330, 4500857724, 16504664528, 60713760264, 223985185896, 828514954047, 3072130220310, 11417124679980, 42518719357968, 158652141816560

Offset

0,3

Links

Table of n, a(n) for n=0..28.

Formula

a(n) = (-1)^n*A223027(n+1)/2.

G.f.: Q(x)*C(Q(x))/x where Q(x) = Sum_{n>=1} -(-x)^(n^2) and C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 112*x^6 + 360*x^7 +...
where
1 - x*A(x) + x^2*A(x)^2 = 1 - x + x^4 - x^9 + x^16 - x^25 + x^36 - x^49 +-...

Prog

(PARI) {a(n)=local(Q=sum(k=1, sqrtint(n+1), (-x)^(k^2))+x^2*O(x^n)); polcoeff( (1-sqrt(1+4*Q))/(2*x), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A223027.

Sequence in context: A149843 A226022 A272463 * A291190 A273955 A054542

Adjacent sequences: A217696 A217697 A217698 * A217700 A217701 A217702

Keyword

nonn

Author

Paul D. Hanna, Mar 19 2013

Status

approved