A122448 Diagonal elements A122445(n+1,n) of the pendular trinomial triangle A122445.

1, 1, 3, 10, 36, 135, 525, 2094, 8524, 35266, 147862, 626884, 2682940, 11575707, 50295809, 219879814, 966487380, 4268781902, 18936044682, 84326759820, 376853237480, 1689551241606, 7597003401186, 34251504489484

Offset

0,3

Links

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

Formula

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

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

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

D-finite with recurrence 4*(n+2)*a(n) +(-9*n-2)*a(n-1) +(-41*n+34)*a(n-2) +2*(-20*n+39)*a(n-3) +4*(n-7)*a(n-4) +4*(7*n-36)*a(n-5) +12*(n-6)*a(n-6)=0. - R. J. Mathar, Feb 06 2025

Mathematica

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

Crossrefs

Cf. A122445, A122446, A122447, A122449, A122450, A122451, A122452.

Sequence in context: A081909 A371873 A126189 * A007582 A369436 A026854

Adjacent sequences: A122445 A122446 A122447 * A122449 A122450 A122451

Keyword

nonn

Author

Paul D. Hanna, Sep 07 2006

Status

approved