A383439 a(n) = (5*n)!/((n!)^2*(3*n + 1)!).

1, 5, 180, 10010, 678300, 51482970, 4206302100, 361913666400, 32356261929420, 2979510862285100, 280884023785324960, 26990111025198348300, 2634899457411931245900, 260690108634780944767200, 26088052554768282442056000, 2636591265602354831196585600, 268771779551047800167424355500

Offset: 0

Author

Peter Luschny, May 03 2025

Keywords

• nonn

Links

• N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025), p. 12.

Crossrefs

Cf. A104978, A383440.

Programs

Maple

a := n -> (5*n)!/((n!)^2*(3*n + 1)!):

Mathematica

Array[(5*#)!/((#!)^2*(3*# + 1)!) &, 17, 0] (* Michael De Vlieger, May 03 2025 *)

Formula

a(n) = A104978(2*n, n), main diagonal of the Bi-Tri array C[m_2, m_3] in the terminology of Wildberger-Rubine.

a(n) ~ 4*(20*n - 3)*3^(5/2-3*n)*5^(5*n-1/2)/(n*(25 + 936*n + 2592*n^2)*Pi). - Stefano Spezia, May 03 2025