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

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, 2014 and J. Int. Seq. 18 (2015) # 15.3.3

Formula

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;

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

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 *)

Prog

(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 */

Crossrefs

Equals 2^n * A048779(n), n > 1.

Sequence in context: A048348 A227384 A199755 * A365766 A297705 A231690

Adjacent sequences: A025746 A025747 A025748 * A025750 A025751 A025752

Keyword

nonn

Author

Olivier Gérard

Status

approved