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A342912 a(n) = [x^n] (1 - 2*x - sqrt((1 - 3*x)/(1 + x)))/(2*x^3).

%I A342912 #20 Jan 05 2025 15:48:04
%S A342912 1,1,3,6,15,36,91,232,603,1585,4213,11298,30537,83097,227475,625992,
%T A342912 1730787,4805595,13393689,37458330,105089229,295673994,834086421,
%U A342912 2358641376,6684761125,18985057351,54022715451,154000562758,439742222071,1257643249140,3602118427251
%N A342912 a(n) = [x^n] (1 - 2*x - sqrt((1 - 3*x)/(1 + x)))/(2*x^3).
%F A342912 D-finite with recurrence a(n) = (2*a(n - 1) + 3*a(n - 2))*(n + 1)/(n + 3) for n >= 3.
%F A342912 a(n) = (-1)^n*hypergeom([1/2, -2 - n], [2], 4).
%F A342912 a(n) ~ (3^(n + 7/2)*(16*n + 11))/(128*sqrt(Pi)*(n + 2)^(5/2)).
%F A342912 G.f.: (M(x) - 1) / (x + x^2) where M(x) is the g.f. of A001006. - _Werner Schulte_, Jan 05 2025
%p A342912 gf := (1 - 2*x - sqrt((1 - 3*x)/(1 + x)))/(2*x^3): ser := series(gf, x, 36):
%p A342912 seq(coeff(ser, x, n), n = 0..30);
%p A342912 a := proc(n) option remember; `if`(n < 3, [1, 1, 3][n + 1],
%p A342912 ((2*a(n - 1) + 3*a(n - 2))*(n + 1))/(n + 3)) end: seq(a(n), n=0..30);
%t A342912 a[n_] := (-1)^n*HypergeometricPFQ[{1/2, -2 - n}, {2}, 4]
%t A342912 Table[a[n], {n, 0, 30}]
%o A342912 (Python)
%o A342912 def rnum():
%o A342912     a, b, n = 1, 3, 3
%o A342912     yield 1
%o A342912     yield 1
%o A342912     while True:
%o A342912         yield b
%o A342912         n += 1
%o A342912         a, b = b, (n*(3*a + 2*b))//(n + 2)
%o A342912 A342912 = rnum()
%o A342912 print([next(A342912) for _ in range(31)])
%Y A342912 The diagonal sums of the Motzkin triangle A064189 (with the Motzkin numbers A001006 as first column), the row sums of A020474, and a shifted version of the Riordan numbers A005043.
%K A342912 nonn,easy
%O A342912 0,3
%A A342912 _Peter Luschny_, Apr 18 2021