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

Offset

0,3

Links

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

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

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

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

Formula

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

a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2; 4*A034301(n-1)= (5*n-6)(!^5) := product(5*j-6, j=2..n). - 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

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 *)

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 */

Crossrefs

Cf. A034687, A049393.

Sequence in context: A116156 A224124 A212472 * A365622 A034325 A335800

Adjacent sequences: A025747 A025748 A025749 * A025751 A025752 A025753

Keyword

nonn

Author

Olivier Gérard

Status

approved