A122447 Central terms of pendular trinomial triangle A122445.

1, 0, 1, 2, 8, 28, 107, 418, 1676, 6848, 28418, 119444, 507440, 2175500, 9400207, 40895602, 178984212, 787503168, 3481278734, 15454765948, 68871993872, 307981243608, 1381569997998, 6215433403188, 28036071086296

Offset: 0

Author

Paul D. Hanna, Sep 07 2006

Keywords

• nonn

Comments

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 A122446, H(x) is g.f. of A122448, F(x) is g.f. of A122450.

Links

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

Crossrefs

Cf. A122445, A122446, A122448, A122449, A122450, A122451, A122452.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) )); // G. C. Greubel, Mar 17 2021

Mathematica

f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
CoefficientList[Series[(1+6*x+2*x^2-f[x])/(2*x*(4+3*x)), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)

PARI

{a(n)=polcoeff(2*(1+2*x)/(1+6*x+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))),n)}

Sage

def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
def A122447_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) ).list()
A122447_list(30) # G. C. Greubel, Mar 17 2021

Formula

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

G.f.: A(x) = ( 1 +6*x +2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4) )/( 2*x*(4+3*x) ).