A119375 Diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.

1, 3, 11, 40, 149, 564, 2166, 8420, 33074, 131085, 523599, 2105727, 8519469, 34652696, 141621164, 581266730, 2394961851, 9902433681, 41074316737, 170869972460, 712729001716, 2980264528670, 12490379959184, 52458339164169

Offset

0,2

Links

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

Formula

G.f.: A(x) = B(x)*(G(x) - 1)/x^2 = B(x)*(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.: (1-2*x-x^2 -sqrt(1-4*x-2*x^2+x^4))/( x^2*(1+2*x^3+x^4 +(1+x)^2*sqrt(1-4*x-2*x^2+x^4)) ). - G. C. Greubel, Mar 16 2021

Mathematica

CoefficientList[Series[(1-2*x-x^2 -Sqrt[1-4*x-2*x^2+x^4])/(x^2*(1+2*x^3+x^4 +(1+x)^2*Sqrt[1-4*x-2*x^2+x^4])), {x, 0, 30}], x] (* G. C. Greubel, Mar 16 2021 *)

Prog

(PARI) {a(n)=polcoeff(2/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))* (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 A119375_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( (1-2*x-x^2 -sqrt(1-4*x-2*x^2+x^4))/( x^2*(1+2*x^3+x^4 +(1+x)^2*sqrt(1-4*x-2*x^2+x^4)) ) ).list()
A119375_list(30) # G. C. Greubel, Mar 16 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-2*x-x^2 - Sqrt(1-4*x-2*x^2+x^4))/( 1+2*x^3+x^4 +(1+x)^2*Sqrt(1-4*x-2*x^2+x^4) ) )); // G. C. Greubel, Mar 16 2021

Crossrefs

Cf. A119369, A119370, A119371, A119372, A119373, A119374, A119376.

Sequence in context: A014301 A346194 A346317 * A149063 A242467 A149064

Adjacent sequences: A119372 A119373 A119374 * A119376 A119377 A119378

Keyword

nonn

Author

Paul D. Hanna, May 17 2006

Status

approved