A330800 Evaluation of the Motzkin polynomials at -1/2 and normalized with (-2)^n.

1, -1, 5, -17, 77, -345, 1653, -8097, 40733, -208553, 1084421, -5708785, 30370861, -163019641, 881790357, -4801746753, 26302052925, -144825094473, 801155664933, -4450426297233, 24815385947469, -138842668857369, 779247587235765, -4385948395419873, 24750623835149661

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*A201641(n,k).

a(n) = (-2)^n*Sum_{k=0..n} A064189(n,k)/(-2)^k.

a(n) = (-36*(n-2)*a(n-3) + 6*(4*n-5)*a(n-2) - (n-5)*a(n-1))/(n+1).

a(n) = [x^n] 2/(sqrt(4*x - 12*x^2 + 1) + 1).

a(n) = [x^n] reverse((x^2 + x)/(3*x^2 + 1))/x.

Maple

a := proc(n) option remember; if n < 3 then return [1, -1, 5][n+1] fi;
(-36*(n - 2)*a(n-3) + 6*(4*n - 5)*a(n-2) - (n - 5)*a(n-1))/(n + 1) end:
seq(a(n), n=0..24);
# Alternative:
gf := 2/(sqrt(4*x - 12*x^2 + 1) + 1):
ser := series(gf, x, 30): seq(coeff(ser, x, n), n=0..24);
# Or:
series((x^2+x)/(3*x^2+1), x, 30): gfun:-seriestoseries(%, 'revogf'):
convert(%, polynom) / x: seq(coeff(%, x, n), n=0..24);

Mathematica

A330800[n_]:= Coefficient[Series[2/(Sqrt[4*x-12*x^2+1] +1), {x, 0, 50}], x, n]; Table[A330800[n], {n, 0, 30}] (* G. C. Greubel, Sep 13 2023 *)

Prog

(SageMath)
R.<x> = PowerSeriesRing(QQ)
f = (x^2 + x)/(3*x^2 + 1)
f.reverse().shift(-1).list()
(Magma) I:=[1, -1, 5]; [n le 3 select I[n] else ((6-n)*Self(n-1) + 6*(4*n-9)*Self(n-2) -36*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Sep 13 2023

Crossrefs

Cf. A064189, A129400, A201641, A330799.

Sequence in context: A149743 A323794 A151479 * A293458 A009234 A211474

Adjacent sequences: A330797 A330798 A330799 * A330801 A330802 A330803

Keyword

sign

Author

Peter Luschny, Jan 01 2020

Status

approved