A025750 5th-order Patalan numbers (generalization of Catalan numbers).

1, 1, 10, 150, 2625, 49875, 997500, 20662500, 439078125, 9513359375, 209293906250, 4661546093750, 104884787109375, 2380077861328125, 54401779687500000, 1251240932812500000, 28934946571289062500, 672311993862304687500, 15687279856787109375000, 367412607172119140625000

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.: (6-(1-25*x)^(1/5))/5.

a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2, where 4*A034301(n-1) = (5*n-6)(!^5) = Product_{j=2..n} (5*j-6). - Wolfdieter Lang

a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n+k-1,n-1) * Sum_{j=0..k} 2^j*binomial(k,j) * Sum_{i=j..n-k+j-1} binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i)/n, n > 0, a(0) = 1. - Vladimir Kruchinin, Dec 10 2011

a(n) = ((-5)^(n-1)*Sum_{k=1..n} (5)^(n-k)*stirling1(n,k))/n!, n>0, a(0) = 1. - Vladimir Kruchinin, Mar 19 2013

From Karol A. Penson, Feb 05 2025: (Start)

a(n) without the initial 1 (i.e., a(n) for n >= 1) is given by

a(n+1) = 5^(2*n)*Gamma(n + 4/5)/(Gamma(4/5)*(n + 1)!), n >= 0.

a(n+1) = Integral_{x=0..25} x^n*W(x) dx, n >= 0,

where W(x) = sin(Pi/5)*5^(2/5)*(1 - x/25)^(1/5)/(5*Pi*x^(1/5)). W(x) is a positive function on x = (0, 25), is singular at x = 0 with the singularity (x)^(-1/5), and it goes to zero at x = 25. (End)

a(n) ~ 25^(n-1) / (Gamma(4/5) * n^(6/5)). - Amiram Eldar, Aug 20 2025

Mathematica

CoefficientList[Series[(6-(1-25x)^(1/5))/5, {x, 0, 20}], x] (* Harvey P. Dale, Dec 06 2012 *)
a[0] = 1; a[n_] := ((-5)^(n - 1)*Sum[5^(n - k)*StirlingS1[n, k], {k, 1, n}])/n!; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 19 2013, after Vladimir Kruchinin *)
a[n_] := 25^(n-1) * Pochhammer[4/5, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

Prog

(Maxima)
a(n):=if n=0 then 1 else (sum((-1)^(n-k-1)*binomial(n+k-1, n-1)*sum(2^j*binomial(k, j)*sum(binomial(j, i-j)*binomial(k-j, n-3*(k-j)-i-1)*5^(3*(k-j)+i), i, j, n-k+j-1), j, 0, k), k, 0, n-1))/(n); /* Vladimir Kruchinin, Dec 10 2011 */
(Maxima)
a(n):=if n=0 then 1 else -binomial(1/5, n)*(-25)^n/5; /* Tani Akinari, Sep 17 2015 */
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( (6 - (1-5^2*x)^(1/5))/5 )); // G. C. Greubel, Oct 29 2025
(SageMath)
def A025750(n): return 1 if n==0 else 5^(2*n-2)*rising_factorial(4/5, n-1)/factorial(n)
[A025750(n) for n in range(41)] # G. C. Greubel, Oct 29 2025

Crossrefs

Cf. A034301, A034687, A049393, A340722.

Sequence in context: A116156 A224124 A212472 * A365622 A034325 A335800

Adjacent sequences: A025747 A025748 A025749 * A025751 A025752 A025753

Keyword

nonn

Author

Olivier Gérard

Status

approved