A122448 Diagonal elements A122445(n+1,n) of the pendular trinomial triangle A122445.

1, 1, 3, 10, 36, 135, 525, 2094, 8524, 35266, 147862, 626884, 2682940, 11575707, 50295809, 219879814, 966487380, 4268781902, 18936044682, 84326759820, 376853237480, 1689551241606, 7597003401186, 34251504489484

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, 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!( 2/(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-x+2*x^2+2*x^3 +(1+x)*f[x]), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)

PARI

{a(n) =polcoeff(2/(1-x+2*x^2+2*x^3 +(1+x)*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( 2/(1-x+2*x^2+2*x^3 +(1+x)*f(x)) ).list()
A122447_list(30) # G. C. Greubel, Mar 17 2021

Formula

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

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

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

D-finite with recurrence 4*(n+2)*a(n) +(-9*n-2)*a(n-1) +(-41*n+34)*a(n-2) +2*(-20*n+39)*a(n-3) +4*(n-7)*a(n-4) +4*(7*n-36)*a(n-5) +12*(n-6)*a(n-6)=0. - R. J. Mathar, Feb 06 2025