A344506 a(n) = [x^n] 2 / (1 - 7*x + sqrt(1 - 2*x - 3*x^2)).

1, 4, 17, 73, 315, 1362, 5895, 25528, 110579, 479068, 2075683, 8993897, 38971621, 168871854, 731764089, 3170939841, 13740635787, 59542470588, 258016586955, 1118069698011, 4844962624953, 20994821090790, 90977510544237, 394235745437286, 1708354520308101

Offset: 0

Author

Peter Luschny, May 23 2021

Keywords

• nonn

Comments

The Motzkin polynomials (coefficients in A064189) evaluated at x = 3.

Crossrefs

The Motzkin polynomials evaluated at: x = 0 (A001006), x = 1 (A005773), x = 2 (A059738), x = 3 (this sequence).

Cf. A064189, A344507.

Programs

Maple

gf := 2 / (1 - 7*x + sqrt(1 - 2*x - 3*x^2)):
ser := series(gf, x, 27): seq(coeff(ser, x, n), n=0..25);
# Or:
rgf := (3*x^2 + x)/(13*x^2 + 7*x + 1):
subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 28), 'revogf'));

Mathematica

RecurrenceTable[{a[n] == (39 (2 - n) a[n - 3] - (17 n + 5) a[n - 2] + (19 n + 10) a[n - 1])/(3 n + 3), a[0] == 1, a[1] == 4, a[2] == 17}, a, {n, 0, 26}]

SageMath

R. = PowerSeriesRing(QQ, default_prec=25)
f = (3*x^2 + x) / (13*x^2 + 7*x + 1)
f.reverse().shift(-1).list()

Formula

a(n) = [x^n] reverse((3*x^2 + x) / (13*x^2 + 7*x + 1)) / x.

a(n) = Sum_{k=0..n} 3^k*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).

a(n) = (39*(2 - n)*a(n - 3) - (17*n + 5)*a(n - 2) + (19*n + 10)*a(n - 1))/(3*n + 3) for n >= 3.

a(n) ~ 8 * 13^n / 3^(n+2). - Vaclav Kotesovec, May 24 2021

G.f.: 1/(1 - 4*x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021