This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A344506 #21 Nov 19 2021 11:10:47
%S A344506 1,4,17,73,315,1362,5895,25528,110579,479068,2075683,8993897,38971621,
%T A344506 168871854,731764089,3170939841,13740635787,59542470588,258016586955,
%U A344506 1118069698011,4844962624953,20994821090790,90977510544237,394235745437286,1708354520308101
%N A344506 a(n) = [x^n] 2 / (1 - 7*x + sqrt(1 - 2*x - 3*x^2)).
%C A344506 The Motzkin polynomials (coefficients in A064189) evaluated at x = 3.
%F A344506 a(n) = [x^n] reverse((3*x^2 + x) / (13*x^2 + 7*x + 1)) / x.
%F A344506 a(n) = Sum_{k=0..n} 3^k*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
%F A344506 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 A344506 a(n) ~ 8 * 13^n / 3^(n+2). - _Vaclav Kotesovec_, May 24 2021
%F A344506 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 A344506 gf := 2 / (1 - 7*x + sqrt(1 - 2*x - 3*x^2)):
%p A344506 ser := series(gf, x, 27): seq(coeff(ser, x, n), n=0..25);
%p A344506 # Or:
%p A344506 rgf := (3*x^2 + x)/(13*x^2 + 7*x + 1):
%p A344506 subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 28), 'revogf'));
%t A344506 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 A344506 (SageMath)
%o A344506 R.<x> = PowerSeriesRing(QQ, default_prec=25)
%o A344506 f = (3*x^2 + x) / (13*x^2 + 7*x + 1)
%o A344506 f.reverse().shift(-1).list()
%Y A344506 The Motzkin polynomials evaluated at: x = 0 (A001006), x = 1 (A005773), x = 2 (A059738), x = 3 (this sequence).
%Y A344506 Cf. A064189, A344507.
%K A344506 nonn
%O A344506 0,2
%A A344506 _Peter Luschny_, May 23 2021