A364402 - OEIS

%I #31 Sep 24 2023 09:15:50

%S 1,126,41990,15967980,6421422150,2663825039876,1127155102890908,

%T 483537022180231320,209536624110664757830,91505601042318156186900,

%U 40205863224219682380130740,17753412284992688334256754280,7871411119532225034145860092700,3502017467737750755575471520717480

%N a(n) = (3*n)!*(10*n)!/((2*n)!*(5*n)!*(6*n)!).

%C Member of Bober's second infinite family of integral factorial ratio sequences with a=5 and b=3 (see equation 11 at p. 16 in Bober).

%H Jonathan W. Bober, <a href="https://doi.org/10.1112/jlms/jdn078">Factorial ratios, hypergeometric series, and a family of step functions</a>, Journal of the London Mathematical Society, Vol. 79, No. 2 (2009), pp. 422-444; <a href="https://arxiv.org/abs/0709.1977">arXiv preprint</a>, arXiv:0709.1977 [math.NT], 2007.

%F a(n) = 10*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)/(3*n*(2*n - 1)*(6*n - 1)*(6*n - 5))*a(n-1).

%F a(n) ~ 2^(2*n-1) * 5^(5*n) / (sqrt(Pi*n) * 3^(3*n)). - _Vaclav Kotesovec_, Sep 21 2023

%F From _Peter Bala_, Sep 24 2023: (Start)

%F a(n) = A262732(2*n).

%F a(n) = [x^(2*n)] (1 + 4*x)^((10*n-1)/2) = 16^n * binomial((10*n-1)/2, 2*n).

%F O.g.f. A(x) = hypergeom([9/10, 7/10, 3/10, 1/10], [5/6, 1/2, 1/6], (12500/27)*x).

%F (End)

%p seq( (3*n)!*(10*n)!/((2*n)!*(5*n)!*(6*n)!), n = 0..20); # _Peter Bala_, Sep 24 2023

%o (PARI) a(n) = (3*n)!*(10*n)!/((2*n)!*(5*n)!*(6*n)!); \\ _Michel Marcus_, Sep 20 2023

%Y Bisection of A262732. Cf. A182400, A211419.

%K nonn

%O 0,2

%A _Neven Sajko_, Jul 22 2023