A182959 G.f. 2*(1+x)^2/(1-2*x+sqrt(1-8*x)).

1, 5, 20, 96, 528, 3136, 19584, 126720, 841984, 5710848, 39376896, 275185664, 1944821760, 13875707904, 99807723520, 722997411840, 5269761884160, 38620004352000, 284405842575360, 2103530005463040, 15619068033761280

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

Let F(x) be the g.f. of A182960, then g.f. of this sequence satisfies:

* A(x) = F(x/A(x)^3) and A(x*F(x)^3) = F(x);

* A(x) = [x/Series_Reversion( x*F(x)^3 )]^(1/3).

G.f.: 1/2/x - 1/2 - x - (1+x)/x/G(0), where G(k)= 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

a(n) ~ 9*2^(3*n-2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013

From Peter Bala, Oct 04 2015: (Start)

O.g.f. A(x) = (1 + x)*(2*C(2*x) - 1), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.

[x^n] A(x)^(3*n) = binomial(6*n,2*n). Cf. with the identity [x^n] ( (1 + x)*C(x) )^(5*n) = binomial(5*n,2*n) = A001450(n). (End)

Conjecture: D-finite with recurrence (n+1)*a(n) +(-7*n+3)*a(n-1) +4*(-2*n+5)*a(n-2)=0. - R. J. Mathar, Jan 22 2020

From Peter Bala, May 15 2023: (Start)

a(n) = 3*(2^n)*(3*n - 1)/(n*(n + 1)) * binomial(2*n-2,n-1) for n >= 2.

(n + 1)*(3*n - 4)*a(n) = 4*(2*n - 3)*(3*n - 1)*a(n-1) for n >= 3 with a(2) = 20. Mathar's conjectured second order recurrence above follows from this. (End)

Example

G.f.: A(x) = 1 + 5*x + 20*x^2 + 96*x^3 + 528*x^4 + 3136*x^5 +...
where A(x*F(x)^3) = F(x) is the g.f. of A182960:
F(x) = 1 + 5*x + 95*x^2 + 2496*x^3 + 76063*x^4 + 2524161*x^5 +...

Mathematica

CoefficientList[ Series[2 (1 + x)^2/(1 - 2 x + Sqrt[1 - 8 x]), {x, 0, 20}], x]  (* Robert G. Wilson v, Dec 31 2010 *)

Prog

(PARI) {a(n)=polcoeff(2*(1+x)^2/(1-2*x+sqrt(1-8*x+x*O(x^n))), n)}

Crossrefs

Cf. A182960, A001450, A167422.

Sequence in context: A352149 A196532 A002745 * A224661 A357786 A020046

Adjacent sequences: A182956 A182957 A182958 * A182960 A182961 A182962

Keyword

nonn,easy

Author

Paul D. Hanna, Dec 31 2010

Status

approved