A187594 G.f. satisfies: 1/A(x) = Sum_{n>=0} (n+1)^2*A000108(n)*x^n*A(x)^n, where A000108(n) = C(2n,n)/(n+1) is the n-th Catalan number.

1, -4, 14, -40, 78, 0, -836, 4224, -11050, 0, 162564, -905216, 2553740, 0, -41941512, 243433472, -711366810, 0, 12363548340, -73437020160, 219033228260, 0, -3940756420536, 23755279564800, -71807057068292, 0, 1322443755038504

Offset

0,2

Links

Table of n, a(n) for n=0..26.

Formula

a(4n+5) = 0 for n>=0.

G.f. satisfies:

(1) [x^(2n-1)] 1/A(x)^n = 0 for n>=2.

(2) A(x) = (1 - 4*x*A(x))^(3/2) / (1 - 2*x*A(x)).

(3) A(x) = (1/x)*Series_Reversion(x*(1-2*x)/(1-4*x)^(3/2)).

(4) (1/x)*Series_Reversion(x*sqrt(1-4*x*A(x))) = sqrt(sqrt(1+16*x^2) + 4*x).

Example

G.f.: A(x) =  1 - 4*x + 14*x^2 - 40*x^3 + 78*x^4 - 836*x^6 +...
where A(x) satisfies:
1/A(x) = 1 + 4*x*A(x) + 18*x^2*A(x)^2 + 80*x^3*A(x)^3 + 350*x^4*A(x)^4 +...+ (n+1)^2*A000108(n)*x^n*A(x)^n +...
Related expansion.
1/A(x) = 1 + 4*x + 2*x^2 - 8*x^3 + 22*x^4 - 32*x^5 - 76*x^6 +...
The coefficients in 1/A(x)^n begin:
n=1: [1,4,2,-8,22,-32,-76,768,-3034,5632,13372,-159744,692476,...];
n=2: [1,8,20, 0 ,-16,80,-256,448,768,-10128,45056,-102784,...];
n=3: [1,12,54,88,-18, 0 ,140,-768,2646,-5120,-6732,110592,...];
n=4: [1,16,104,320,368,-96,128, 0 ,-1280,7392,-26624,54912,...];
n=5: [1,20,170,760,1750,1504,-380,640,-1050, 0 ,12012,-71680,...];
n=6: [1,24,252,1472,4992,9072,6080,-1344,2304,-5040,9216, 0 ,...]; ...
where the coefficient of x^(2n-1) in 1/A(x)^n equals zero for n>=2.
...
Let G(x) = (1 - 4*x*A(x))^(1/2) = [A(x)*(1 - 2*x*A(x))]^(1/3), where:
G(x) = 1 - 2*x + 6*x^2 - 16*x^3 + 30*x^4 - 308*x^6 + 1536*x^7 +...
then (1/x)*Series_Reversion(x*G(x)) = F(x), where:
F(x) = 1 + 2*x + 2*x^2 - 4*x^3 - 10*x^4 + 28*x^5 + 84*x^6 - 264*x^7 - 858*x^8 +...+ (-1)^[n/2]*2*A000108(n)*x^(n+1) +...;
F(x) = sqrt(sqrt(1+16*x^2) + 4*x) = [(1-I)*sqrt(1+4*I*x)+(1+I)*sqrt(1-4*I*x)]/2.

Prog

(PARI) {a(n)=polcoeff(serreverse(x*(1-2*x)/(1-4*x+x*O(x^n))^(3/2))/x, n)}
(PARI) {a(n)=local(C=sum(m=0, n, (m+1)*binomial(2*m, m)*x^m)+x*O(x^n)); polcoeff(serreverse(x*C)/x, n)}

Crossrefs

Cf. A000108.

Sequence in context: A074083 A182819 A144141 * A326482 A331758 A291675

Adjacent sequences: A187591 A187592 A187593 * A187595 A187596 A187597

Keyword

sign

Author

Paul D. Hanna, Mar 11 2011

Status

approved