A371986 Product of Lucas and Catalan numbers: a(n) = A000032(n)*A000108(n).

2, 1, 6, 20, 98, 462, 2376, 12441, 67210, 369512, 2065908, 11698414, 66979864, 387050900, 2254552920, 13223768580, 78034377690, 462961545090, 2759796408600, 16522143563310, 99295449593340, 598836351581520, 3622983967834920, 21982916983078350, 133739841802846968

Offset: 0

Author

Vladimir Kruchinin, Apr 15 2024

Keywords

• nonn

Crossrefs

Cf. A000032, A000108, A098614, A119694, A127211, A216541.

Programs

Maple

From Peter Luschny, Apr 15 2024: (Start)
a := n -> ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * GAMMA(n + 1/2) / (sqrt(Pi) * GAMMA(n + 2)): seq(simplify(a(n)), n = 0..24);
# With g.f.:
assume(x>0); f := sqrt(1 - 4*x*(4*x + 1)):
gf := (sqrt(1 + f - 2*x) + sqrt(5)*sqrt(1 - f - 2*x) - sqrt(2))/(sqrt(8)*x):
ser := series(gf, x, 26): seq(simplify(coeff(ser, x, n)), n = 0..24);
# Recurrence:
a := proc(n) option remember: if n < 2 then return [2, 1][n + 1] fi;
2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) end:
seq(a(n), n=0..24);  (End)

Mathematica

a[n_] := a[n] = Switch[n, 0, 2, 1, 1, _, 2*(2n - 1)*(n*a[n - 1] + (4n - 6)*a[n - 2])/(n*(n + 1))];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 17 2024, after Peter Luschny *)

Python

def A371986_gen(): # generator of terms
    a, b, n = 2, 1, 2
    while True:
        yield a
        a, b = b, (4*n - 2)*(n*b + (4*n - 6)*a) // (n*n + n)
        n += 1
def A371986_list(len):
    it =  A371986_gen()
    return [next(it) for _ in range(len)]
print(A371986_list(25))  # Peter Luschny, Apr 15 2024

Formula

G.f.: (5*sqrt(-sqrt(-16*x^2 - 4*x+1) - 2*x+1)) / (2*sqrt(10)*x) - (1 - sqrt(sqrt( -16*x^2 - 4*x+1) - 2*x + 1) / sqrt(2)) / (2*x).

E.g.f.: exp(x-sqrt(5)*x)*(BesselI(0, x-sqrt(5)*x) - BesselI(1, x-sqrt(5)*x) + exp(2*sqrt(5)*x) * (BesselI(0, x+sqrt(5)*x) - BesselI(1, x+sqrt(5)*x))). - Stefano Spezia, Apr 15 2024

From Peter Luschny, Apr 15 2024: (Start)

a(n) = 2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) for n >= 2.

a(n) = ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * Gamma(n + 1/2) / (sqrt(Pi) * Gamma(n + 2)).

a(n) ~ (2 + 2*sqrt(5))^n / (n*(n*Pi)^(1/2)). (End)