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

%I #21 Nov 19 2021 11:10:47

%S 1,4,17,73,315,1362,5895,25528,110579,479068,2075683,8993897,38971621,

%T 168871854,731764089,3170939841,13740635787,59542470588,258016586955,

%U 1118069698011,4844962624953,20994821090790,90977510544237,394235745437286,1708354520308101

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

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

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

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

%F 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.

%F a(n) ~ 8 * 13^n / 3^(n+2). - _Vaclav Kotesovec_, May 24 2021

%F 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

%p gf := 2 / (1 - 7*x + sqrt(1 - 2*x - 3*x^2)):

%p ser := series(gf, x, 27): seq(coeff(ser, x, n), n=0..25);

%p # Or:

%p rgf := (3*x^2 + x)/(13*x^2 + 7*x + 1):

%p subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 28), 'revogf'));

%t 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}]

%o (SageMath)

%o R.<x> = PowerSeriesRing(QQ, default_prec=25)

%o f = (3*x^2 + x) / (13*x^2 + 7*x + 1)

%o f.reverse().shift(-1).list()

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

%Y Cf. A064189, A344507.

%K nonn

%O 0,2

%A _Peter Luschny_, May 23 2021