A122447 Central terms of pendular trinomial triangle A122445.

1, 0, 1, 2, 8, 28, 107, 418, 1676, 6848, 28418, 119444, 507440, 2175500, 9400207, 40895602, 178984212, 787503168, 3481278734, 15454765948, 68871993872, 307981243608, 1381569997998, 6215433403188, 28036071086296

Offset

0,4

Comments

G.f.: A(x) = 1/(1+x - x*B(x)) = (1 + x*H(x))/(1+x) = 1 + x^2*F(x)/B(x), where B(x) is g.f. of A122446, H(x) is g.f. of A122448, F(x) is g.f. of A122450.

Links

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

Formula

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

G.f.: A(x) = ( 1 +6*x +2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4) )/( 2*x*(4+3*x) ).

Mathematica

f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
CoefficientList[Series[(1+6*x+2*x^2-f[x])/(2*x*(4+3*x)), {x, 0, 30}], x] (* G. C. Greubel, Mar 17 2021 *)

Prog

(PARI) {a(n)=polcoeff(2*(1+2*x)/(1+6*x+2*x^2+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.<x> = PowerSeriesRing(QQ, prec)
    return P( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) ).list()
A122447_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!( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) )); // G. C. Greubel, Mar 17 2021

Crossrefs

Cf. A122445, A122446, A122448, A122449, A122450, A122451, A122452.

Sequence in context: A357641 A150714 A292668 * A150715 A356933 A026528

Adjacent sequences: A122444 A122445 A122446 * A122448 A122449 A122450

Keyword

nonn

Author

Paul D. Hanna, Sep 07 2006

Status

approved