%I #12 Jun 27 2022 05:24:06
%S 1,1,3,6,15,36,91,232,603,1585,4213,11298,30537,83097,227475,625992,
%T 1730787,4805595,13393689,37458330,105089229,295673994,834086421,
%U 2358641376,6684761125,18985057351,54022715451,154000562758,439742222071,1257643249140,3602118427251
%N a(n) = [x^n] (1 - 2*x - sqrt((1 - 3*x)/(1 + x)))/(2*x^3).
%F D-finite with recurrence a(n) = (2*a(n - 1) + 3*a(n - 2))*(n + 1)/(n + 3) for n >= 3.
%F a(n) = (-1)^n*hypergeom([1/2, -2 - n], [2], 4].
%F a(n) ~ (3^(n + 7/2)*(16*n + 11))/(128*sqrt(Pi)*(n + 2)^(5/2)).
%p gf := (1 - 2*x - sqrt((1 - 3*x)/(1 + x)))/(2*x^3): ser := series(gf, x, 36):
%p seq(coeff(ser, x, n), n = 0..30);
%p a := proc(n) option remember; `if`(n < 3, [1, 1, 3][n + 1],
%p ((2*a(n - 1) + 3*a(n - 2))*(n + 1))/(n + 3)) end: seq(a(n), n=0..30);
%t a[n_] := (-1)^n*HypergeometricPFQ[{1/2, -2 - n}, {2}, 4]
%t Table[a[n], {n, 0, 30}]
%o (Python)
%o def rnum():
%o a, b, n = 1, 3, 3
%o yield 1
%o yield 1
%o while True:
%o yield b
%o n += 1
%o a, b = b, (n*(3*a + 2*b))//(n + 2)
%o A342912 = rnum()
%o print([next(A342912) for _ in range(31)])
%Y 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 nonn,easy
%O 0,3
%A _Peter Luschny_, Apr 18 2021