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

1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595, 13393689, 37458330, 105089229, 295673994, 834086421, 2358641376, 6684761125, 18985057351, 54022715451, 154000562758, 439742222071, 1257643249140, 3602118427251

Offset: 0

Author

Peter Luschny, Apr 18 2021

Keywords

• nonn
• easy

Crossrefs

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.

Programs

Maple

gf := (1 - 2*x - sqrt((1 - 3*x)/(1 + x)))/(2*x^3): ser := series(gf, x, 36):
seq(coeff(ser, x, n), n = 0..30);
a := proc(n) option remember; `if`(n < 3, [1, 1, 3][n + 1],
((2*a(n - 1) + 3*a(n - 2))*(n + 1))/(n + 3)) end: seq(a(n), n=0..30);

Mathematica

a[n_] := (-1)^n*HypergeometricPFQ[{1/2, -2 - n}, {2}, 4]
Table[a[n], {n, 0, 30}]

Python

def rnum():
    a, b, n = 1, 3, 3
    yield 1
    yield 1
    while True:
        yield b
        n += 1
        a, b = b, (n*(3*a + 2*b))//(n + 2)
A342912 = rnum()
print([next(A342912) for _ in range(31)])

Formula

D-finite with recurrence a(n) = (2*a(n - 1) + 3*a(n - 2))*(n + 1)/(n + 3) for n >= 3.

a(n) = (-1)^n*hypergeom([1/2, -2 - n], [2], 4).

a(n) ~ (3^(n + 7/2)*(16*n + 11))/(128*sqrt(Pi)*(n + 2)^(5/2)).

G.f.: (M(x) - 1) / (x + x^2) where M(x) is the g.f. of A001006. - Werner Schulte, Jan 05 2025