A025761 8th-order Vatalan numbers (generalization of Catalan numbers).

1, 1, 29, 1177, 54629, 2726977, 142504685, 7685245225, 424109499317, 23818681210961, 1356315674712509, 78100982458201017, 4538960021319997189, 265837773438037013857, 15672475449746510425485

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A025756, A025757, A025758, A025759, A025760, A025762, A025763.

Programs

Mathematica

Table[SeriesCoefficient[8/(7 + (1 - 64 x)^(1/8)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)
CoefficientList[Series[8/(7+Surd[1-64x,8]),{x,0,20}],x] (* Harvey P. Dale, Jun 19 2022 *)

Formula

G.f.: 8/(7 + (1 - 64*x)^(1/8)).

a(n) = (-1)^(n+1) * 8^(2*n+1) * Sum_{k>=0} (-1/7)^(k+1) * binomial(k/8,n). - Seiichi Manyama, Aug 04 2024