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

%I A371986 #29 Jun 17 2024 11:50:34
%S A371986 2,1,6,20,98,462,2376,12441,67210,369512,2065908,11698414,66979864,
%T A371986 387050900,2254552920,13223768580,78034377690,462961545090,
%U A371986 2759796408600,16522143563310,99295449593340,598836351581520,3622983967834920,21982916983078350,133739841802846968
%N A371986 Product of Lucas and Catalan numbers: a(n) = A000032(n)*A000108(n).
%F A371986 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).
%F A371986 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
%F A371986 From _Peter Luschny_, Apr 15 2024: (Start)
%F A371986 a(n) = 2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) for n >= 2.
%F A371986 a(n) = ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * Gamma(n + 1/2) / (sqrt(Pi) * Gamma(n + 2)).
%F A371986 a(n) ~ (2 + 2*sqrt(5))^n / (n*(n*Pi)^(1/2)). (End)
%p A371986 From _Peter Luschny_, Apr 15 2024: (Start)
%p A371986 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);
%p A371986 # With g.f.:
%p A371986 assume(x>0); f := sqrt(1 - 4*x*(4*x + 1)):
%p A371986 gf := (sqrt(1 + f - 2*x) + sqrt(5)*sqrt(1 - f - 2*x) - sqrt(2))/(sqrt(8)*x):
%p A371986 ser := series(gf, x, 26): seq(simplify(coeff(ser, x, n)), n = 0..24);
%p A371986 # Recurrence:
%p A371986 a := proc(n) option remember: if n < 2 then return [2, 1][n + 1] fi;
%p A371986 2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) end:
%p A371986 seq(a(n), n=0..24);  (End)
%t A371986 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))];
%t A371986 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Jun 17 2024, after _Peter Luschny_ *)
%o A371986 (Python)
%o A371986 def A371986_gen(): # generator of terms
%o A371986     a, b, n = 2, 1, 2
%o A371986     while True:
%o A371986         yield a
%o A371986         a, b = b, (4*n - 2)*(n*b + (4*n - 6)*a) // (n*n + n)
%o A371986         n += 1
%o A371986 def A371986_list(len):
%o A371986     it =  A371986_gen()
%o A371986     return [next(it) for _ in range(len)]
%o A371986 print(A371986_list(25))  # _Peter Luschny_, Apr 15 2024
%Y A371986 Cf. A000032, A000108, A098614, A119694, A127211, A216541.
%K A371986 nonn
%O A371986 0,1
%A A371986 _Vladimir Kruchinin_, Apr 15 2024