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

1, -1, 2, -2, 5, -3, 15, 3, 59, 73, 308, 632, 1951, 4829, 13674, 36306, 100827, 275493, 765150, 2120466, 5918943, 16547595, 46452387, 130703031, 368825661, 1043125407, 2957013140, 8399389528, 23904802109, 68154435941, 194639738503, 556733127851, 1594781146419

Offset: 0

Author

Peter Luschny, May 23 2021

Keywords

• sign

Links

• Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.

Crossrefs

Cf. A005043, A001006, A005773, A059738, A344506, A330799.

Programs

Maple

gf := 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1):
ser := series(gf, x, 27): seq(coeff(ser, x, n), n = 0..25);
# Or:
rgf := (x - 2*x^2) / (3*x^2 - 3*x + 1):
subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 32), 'revogf'));

Mathematica

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}]
(* Or: *)
rgf := (x - 2 x^2) / (3 x^2 - 3 x + 1);
CoefficientList[InverseSeries[Series[rgf, {x, 0, 32}]] / x, x]

SageMath

R. = PowerSeriesRing(QQ, default_prec=32)
f = (x - 2*x^2) / (3*x^2 - 3*x + 1)
f.reverse().shift(-1).list()

Formula

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

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

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.