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A344507 a(n) = [x^n] 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1).

%I A344507 #13 Mar 22 2023 18:13:07
%S A344507 1,-1,2,-2,5,-3,15,3,59,73,308,632,1951,4829,13674,36306,100827,
%T A344507 275493,765150,2120466,5918943,16547595,46452387,130703031,368825661,
%U A344507 1043125407,2957013140,8399389528,23904802109,68154435941,194639738503,556733127851,1594781146419
%N A344507 a(n) = [x^n] 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1).
%H A344507 Taras Goy and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Shattuck/sh36.html">Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries</a>, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.
%F A344507 a(n) = [x^n] reverse((x - 2*x^2) / (3*x^2 - 3*x + 1)) / x.
%F A344507 a(n) = Sum_{k=0..n}(-2)^k*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
%F A344507 a(n) = (9*(n - 2)*a(n - 3) + (12*n - 15)*a(n - 2) + (n - 5)*a(n - 1))/(2*n + 2) for n >= 3.
%p A344507 gf := 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1):
%p A344507 ser := series(gf, x, 27): seq(coeff(ser, x, n), n = 0..25);
%p A344507 # Or:
%p A344507 rgf := (x - 2*x^2) / (3*x^2 - 3*x + 1):
%p A344507 subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 32), 'revogf'));
%t A344507 a[n_] := Sum[(-2)^k Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 2, 4], {k, 0, n}]; Table[a[n], {n, 0, 32}]
%t A344507 (* Or: *)
%t A344507 rgf := (x - 2 x^2) / (3 x^2 - 3 x + 1);
%t A344507 CoefficientList[InverseSeries[Series[rgf, {x, 0, 32}]] / x, x]
%o A344507 (SageMath)
%o A344507 R.<x> = PowerSeriesRing(QQ, default_prec=32)
%o A344507 f = (x - 2*x^2) / (3*x^2 - 3*x + 1)
%o A344507 f.reverse().shift(-1).list()
%Y A344507 Cf. A005043, A001006, A005773, A059738, A344506, A330799.
%K A344507 sign
%O A344507 0,3
%A A344507 _Peter Luschny_, May 23 2021