A122451 A diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.

1, 4, 17, 72, 305, 1300, 5576, 24068, 104510, 456332, 2002675, 8829892, 39096653, 173781548, 775183764, 3469084436, 15571135682, 70084045640, 316242702258, 1430351652352, 6483550388522, 29448610671464, 134010580021152

Offset

0,2

Links

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

Formula

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

G.f.: 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(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

Mathematica

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

Prog

(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*(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.<x> = PowerSeriesRing(QQ, prec)
    return P( 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(1-x+2*x^2+2*x^3+(1+x)*f(x))) ).list()
A122449_list(30) # G. C. Greubel, Mar 17 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(1-x+2*x^2+2*x^3+(1+x)*f(x))) )); // G. C. Greubel, Mar 17 2021

Crossrefs

Cf. A122445, A122446, A122447, A122448, A122449, A122450, A122452.

Sequence in context: A022031 A255117 A001076 * A257388 A113442 A362908

Adjacent sequences: A122448 A122449 A122450 * A122452 A122453 A122454

Keyword

nonn

Author

Paul D. Hanna, Sep 07 2006

Status

approved