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

0,2

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

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

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 *)

Prog

(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)}
(Sage)
def f(x): return sqrt(1-4*x-2*x^2+x^4)
def A119376_list(prec):
    P.<x> = 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
(Magma)
R<x>:=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

Crossrefs

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

Sequence in context: A077822 A257838 A099503 * A282310 A022030 A135450

Adjacent sequences: A119373 A119374 A119375 * A119377 A119378 A119379

Keyword

nonn

Author

Paul D. Hanna, May 17 2006

Status

approved