A025749 4th-order Patalan numbers (generalization of Catalan numbers).

1, 1, 6, 56, 616, 7392, 93632, 1230592, 16612992, 228890112, 3204461568, 45445091328, 651379642368, 9419951751168, 137262154088448, 2013178259963904, 29694379334467584, 440175505428578304, 6553724191936610304, 97960930026841964544, 1469413950402629468160

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

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.

Crossrefs

Cf. A034176, A048779, A068465.

Programs

Mathematica

a[n_] := (4^(n-1)*Sum[ Binomial[n+k-1, n-1]*Sum[ Binomial[j, n-3*k+2*j-1] * 4^(j-k) * Binomial[k, j] * 3^(-n+3*k-j+1) * 2^(n-3*k+j-1) * (-1)^(n-3*k+2*j-1), {j, 0, k}], {k, 1, n-1}])/n; a[0] = a[1] = 1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Mar 05 2013, after Vladimir Kruchinin *)
a[n_] := 16^(n-1) * Pochhammer[3/4, n-1]/n!; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 20 2025 *)

Maxima

a(n):=(4^(n-1)*sum(binomial(n+k-1,n-1)*sum(binomial(j,n-3*k+2*j-1)*4^(j-k)*binomial(k,j)*3^(-n+3*k-j+1)*2^(n-3*k+j-1)*(-1)^(n-3*k+2*j-1),j,0,k),k,1,n-1))/n; /* Vladimir Kruchinin, Apr 01 2011 */

Formula

a(n) = 2^(n-1) * A048779(n), n > 1.

From Wolfdieter Lang: (Start)

G.f.: (5-(1-16*x)^(1/4))/4.

a(n) = 4^(n-1)*3*A034176(n-1)/n!, n >= 2, where 3*A034176(n-1) = (4*n-5)(!^4) = Product_{j=2..n} (4*j - 5). (End)

a(n) = (4^(n-1) * Sum_{k=1..n-1} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-3*k+2*j-1)*4^(j-k)*binomial(k,j)*3^(-n+3*k-j+1)*2^(n-3*k+j-1)*(-1)^(n-3*k+2*j-1))/n. - Vladimir Kruchinin, Apr 01 2011

n*a(n) + 4*(-4*n+5)*a(n-1) = 0. - R. J. Mathar, Apr 05 2018

a(n) ~ 16^(n-1) / (Gamma(3/4) * n^(5/4)). - Amiram Eldar, Aug 20 2025