A383440 a(n) = (5*n + 3)!/((8*n^2 + 10*n + 3)*(n!)^2*(3*n + 2)!).

1, 16, 702, 42432, 3010700, 235282320, 19615029280, 1712821144320, 154870831986156, 14388837044278400, 1366276815032189060, 132069279628944665280, 12957831870375876372252, 1287484157116598357029120, 129316124278441748161584000, 13111175417326191857901849600

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. 17.

Crossrefs

Cf. A383453, A383439.

Programs

Maple

a := n -> ((5*n + 3)!/((8*n^2 + 10*n + 3)*(n!)^2*(3*n + 2)!)):

Mathematica

Array[(5*# + 3)!/((8*#^2 + 10*# + 3)*(#!)^2*(3*# + 2)!) &, 16, 0] (* Michael De Vlieger, May 03 2025 *)

Formula

a(n) = A383453(2*n, n), conjectured by Wildberger-Rubine to be the main diagonal of the Geode Bi-Tri array G[m_2, m_3].

a(n) ~ 3^(-3*n-5/2)*5^(5*n+7/2)/(16*n^2*Pi). - Stefano Spezia, May 03 2025