A119372 G.f. satisfies: A(x) = 1 + x*(1-x-x^2)*A(x) + x^2*(3+2*x)*A(x)^2.

1, 1, 3, 9, 30, 104, 374, 1380, 5197, 19893, 77170, 302716, 1198729, 4785455, 19238706, 77821522, 316506253, 1293489529, 5309112257, 21876225899, 90459484106, 375256749620, 1561259497099, 6513108751281, 27238006266620

Offset

0,3

Comments

Equals diagonal and row sums of pendular trinomial triangle A119369. Also equals convolution of A119370 and A119371 (central terms of A119369).

Links

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

Formula

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

G.f.: A(x) = B(x)/(1+x - x*B(x)) = B(x)*G(x), where B(x) is g.f. of A119370, G(x) is g.f. of A119371.

Recurrence: 3*(n+2)*(2*n-1)*a(n) = (20*n^2 - 6*n - 11)*a(n-1) + (28*n^2 - 18*n + 5)*a(n-2) + (8*n^2-12*n-17)*a(n-3) - 3*(2*n^2 - 9*n + 1)*a(n-4) - 2*(n-5)*(2*n+1)*a(n-5). - Vaclav Kotesovec, Sep 11 2013

a(n) ~ sqrt(-8*z^2-5*z^3+2-5*z)*(4+2*z-z^3)^n*(-18-8*z+4*z^3+z^2)*(-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

Maple

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

Mathematica

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

Prog

(PARI) {a(n)=polcoeff(2/(1-x+x^2+x^3+sqrt((1-x+x^2+x^3)^2-4*x^2*(3+2*x)+x*O(x^n))), n)}
(Sage)
def A119372_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( (1-x+x^2+x^3 -(1+x)*sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)) ).list()
A119372_list(30) # G. C. Greubel, Mar 17 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-x+x^2+x^3 -(1+x)*Sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)) )); // G. C. Greubel, Mar 17 2021

Crossrefs

Cf. A119369, A119370, A119371, A119373, A119374, A119375, A119376.

Sequence in context: A148954 A289602 A148955 * A145268 A148956 A339036

Adjacent sequences: A119369 A119370 A119371 * A119373 A119374 A119375

Keyword

nonn

Author

Paul D. Hanna, May 17 2006

Status

approved