A122449 Diagonal elements A122445(n+2,n) of the pendular trinomial triangle A122445.

1, 2, 6, 22, 83, 324, 1298, 5302, 22002, 92488, 392996, 1685232, 7283511, 31694460, 138746706, 610601374, 2699835614, 11988069480, 53433418716, 238986495540, 1072250526558, 4824638825032, 21765895919444, 98433111857436

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, 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/(1-2*x-2*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*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*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*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^2/(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*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*f(x)) ).list()
A122449_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f.: A(x) = B(x)^2/(1+x -x*B(x)) where B(x) is the g.f. of A122446.

G.f.: 2/(1 -2*x -2*x^2 -2*x^3 +4*x^4 +4*x^5 +(1 +2*x^2 +2*x^3)*f(x)), where f(x) = sqrt(1 -4*x -4*x^2 +4*x^4). - G. C. Greubel, Mar 17 2021

D-finite with recurrence -4*(n+3)*(37*n-56)*a(n) +(33*n^2-357*n+1624)*a(n-1) +4*(547*n^2-620*n-554)*a(n-2) +4*(1142*n^2-2566*n-1613)*a(n-3) +16*(180*n^2-588*n+65)*a(n-4) +4*(-331*n^2+1937*n+1076)*a(n-5) +8*(-320*n^2+2107*n-617)*a(n-6) -48*(19*n-13)*(n-7)*a(n-7)=0. - R. J. Mathar, Feb 06 2025