A119371 G.f. satisfies: A(x) = (1+x) - x*(4+x)*A(x) + x*(3+2*x)*A(x)^2.

1, 0, 1, 2, 7, 23, 81, 293, 1087, 4110, 15783, 61387, 241329, 957400, 3828055, 15410651, 62410871, 254095382, 1039394147, 4269718110, 17606507789, 72852976317, 302403773303, 1258855723796, 5254253027485, 21983753239135

Offset

0,4

Comments

Equals central terms of pendular trinomial triangle A119369.

Links

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

Formula

G.f.: A(x) = (1+4*x+x^2 - sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x)))/(2*x*(3+2*x)).

G.f.: A(x) = 1/(1+x - x*B(x)) = (1 + x*H(x))/(1+x) = 1 + x^2*F(x)/B(x), where B(x) is g.f. of A119370, H(x) is g.f. of A119372, F(x) is g.f. of A119375.

Recurrence: 3*(n+1)*a(n) = 2*(5*n-4)*a(n-1) + 2*(7*n-8)*a(n-2) + 4*(n-2)*a(n-3) - 3*(n-5)*a(n-4) - 2*(n-5)*a(n-5). - Vaclav Kotesovec, Sep 11 2013

a(n) ~ sqrt(-z^2-3*z+1)*(4+2*z-z^3)^(n+1)*(35-8*z^3+12*z^2-2*z) /(242*sqrt(Pi)*n^(3/2)), where z = 1/(2*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) - 1/2*sqrt(8/3-1/3*(280-24*sqrt(129))^(1/3) - 2/3*(35+3*sqrt(129))^(1/3) + 8*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) = 0.225270426... is the root of the equation 1-2*z^2+z^4-4*z=0. - Vaclav Kotesovec, Sep 11 2013

Example

            A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 23*x^5 + 81*x^6 ...;
   -x*(4+x)*A(x) = -4*x -x^2 -4*x^3 -9*x^4 -30*x^5 -99*x^6 - ...;
x*(3+2*x)*A(x)^2 = 3*x +2*x^2 +6*x^3 +16*x^4 +53*x^5 +180*x^6 + ...;

Maple

m:= 30;
S:= series( (1+4*x+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 17 2021

Mathematica

CoefficientList[Series[(1+4*x+x^2-Sqrt[(1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)])/(2*x*(3+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 11 2013 *)

Prog

(PARI) {a(n)=polcoeff((2*(1+x))/(1+4*x+x^2+sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x*O(x^n))), n)}
(Sage)
def A119371_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( (1+4*x+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)) ).list()
A119371_list(30) # G. C. Greubel, Mar 17 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+4*x+x^2 -Sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)) )); // G. C. Greubel, Mar 17 2021

Crossrefs

Cf. A119369, A119370, A119372, A119373, A119374, A119375, A119376.

Sequence in context: A091514 A143629 A176287 * A151290 A346627 A047002

Adjacent sequences: A119368 A119369 A119370 * A119372 A119373 A119374

Keyword

nonn

Author

Paul D. Hanna, May 17 2006

Status

approved