A119376 Second diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.

1, 4, 16, 63, 248, 980, 3894, 15563, 62555, 252789, 1026623, 4188390, 17159382, 70570380, 291253664, 1205935204, 5008047097, 20854723702, 87064706122, 364334839028, 1527943938306, 6420911995109, 27033938458595

Offset: 0

Author

Paul D. Hanna, May 17 2006

Keywords

• nonn

Comments

Equals convolution of A119370 and A119375, which is the prior diagonal above the central terms of triangle A119369.

Links

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

Crossrefs

Cf. A119369, A119370, A119371, A119372, A119373, A119374, A119375.

Programs

Magma

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

Mathematica

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

PARI

{a(n)=polcoeff(4/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))^2* (2*(1+x)/(1+4*x+x^2 + sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x^3*O(x^n)))-1)/x^2,n)}

SageMath

def f(x): return sqrt(1-4*x-2*x^2+x^4)
def A119376_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) ).list()
A119376_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f.: A(x) = B(x)^2*(G(x) - 1)/x^2 = B(x)^2*(B(x) - 1)/(x+x^2 - x^2*B(x)), where B(x) is g.f. of A119370 and G(x) is g.f. of A119371 (central terms of A119369).

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