A025753 8th-order Patalan numbers (generalization of Catalan numbers).

1, 1, 28, 1120, 51520, 2555392, 132880384, 7137574912, 392566620160, 21983730728960, 1248675905404928, 71742106565083136, 4161042180774821888, 243260927491451125760, 14317643160925409116160, 847604475126784219676672, 50432466270043661070761984, 3014081513550844685170245632

Offset

0,3

Links

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

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.

Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Formula

G.f.: (9-(1-64*x)^(1/9))/8.

a(n) = 8^(n-1)*7*A034975(n-1)/n!, n >= 2, where 7*A034975(n-1)= (8*n-9)!^8 = Product_{j=2..n} (8*j - 9). - Wolfdieter Lang

a(n) ~ 64^(n-1) / (Gamma(7/8) * n^(9/8)). - Amiram Eldar, Aug 20 2025

Mathematica

CoefficientList[Series[(9 - (1 - 64*x)^(1/8))/8, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)
a[n_] := 64^(n-1) * Pochhammer[7/8, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( (9 - (1-8^2*x)^(1/8))/8 )); // G. C. Greubel, Oct 30 2025
(SageMath)
def A025753(n): return 1 if n==0 else 8^(2*n-2)*rising_factorial(7/8, n-1)/factorial(n)
[A025753(n) for n in range(41)] # G. C. Greubel, Oct 30 2025

Crossrefs

Cf. A025748, A034975, A203146.

Sequence in context: A365029 A370358 A386273 * A160312 A132503 A092705

Adjacent sequences: A025750 A025751 A025752 * A025754 A025755 A025756

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved