A122450 Diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.

1, 3, 12, 47, 189, 773, 3208, 13478, 57222, 245134, 1058348, 4600571, 20118753, 88450897, 390721560, 1733348234, 7719287578, 34497374034, 154659735720, 695397289078, 3135087583426, 14168892518258, 64181607367952

Offset: 0

Author

Paul D. Hanna, Sep 07 2006

Keywords

• nonn

Links

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

Crossrefs

Cf. A122445, A122446, A122447, A122448, A122449, A122451, A122452.

Programs

Magma

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

Mathematica

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

PARI

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

Sage

def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
def A122449_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))) ).list()
A122449_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f.: A(x) = B(x)*(B(x)-1)/(x*(1+x -x*B(x))) where B(x) is the g.f. of A122446.

G.f.: 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))), where f(x) = sqrt(1 -4*x -4*x^2 +4*x^4). - G. C. Greubel, Mar 17 2021