A025758 5th-order Vatalan numbers (generalization of Catalan numbers).

1, 1, 11, 171, 3056, 58916, 1191376, 24896436, 532911346, 11617952106, 256966100966, 5750337968926, 129926216608236, 2959472057112396, 67877180959091156, 1566072624078270516, 36319953436423545851

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.

Formula

G.f.: 5 / (4+(1-25*x)^(1/5)).

a(n) = sum(m=1..n-1, 5^(n-m)*m/n * sum(k=1..n-m, binomial(n+k-1,n-1) * sum(i=0..k, binomial(k,i) * 2^(k-i) * sum(j=0..i, binomial(j,-3*i+n-m-k+2*j) * (-1)^(j-i)*5^(j-i)*(-2)^(3*i-n+m+k-j) * binomial(i,j)))))+1. - Vladimir Kruchinin, Feb 09 2011

Conjecture: 41*n*(n-1)*(n-2)*(n-3)*a(n) -3*(1367*n-4100)*(n-1)*(n-2)*(n-3)*a(n-1) +50*(n-2)*(n-3)*(3077*n^2-21533*n+38136)*a(n-2) -250*(n-3)*(10265*n^3-123135*n^2+494446*n-664572)*a(n-3) +1875*(8575*n^4-154250*n^3+1039765*n^2-3112730*n+3491808)*a(n-4) -625*(5*n-22)*(5*n-21)*(5*n-24)*(5*n-23)*a(n-5)=0. - R. J. Mathar, Jul 28 2014

Maple

# Based on Tani Akinari's formula.
h := (n, j) -> ((-1)^n/(-4)^j)*binomial(j/5, n+1)*hypergeom([1, n+1-j/5], [n+2], 1025): a := n -> 2^8*5^(2*n+1)*add(h(n, j), j=1..4):
seq(round(evalf(a(n), 64)), n=0..16); # Peter Luschny, Sep 21 2015

Mathematica

Table[SeriesCoefficient[5/(4 + (1 - 25*x)^(1/5)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

Prog

(Maxima) a(n):=(256/205)*41^(-n)*sum(sum((-4)^(-k)*(-1025)^m*binomial(k/5, m), k, 0, 4), m, 0, n);  /* Tani Akinari, Sep 16 2015  */

Crossrefs

a(n), n >= 1, = row sums of triangle A049223.

Sequence in context: A205087 A064182 A139792 * A243677 A307168 A141955

Adjacent sequences: A025755 A025756 A025757 * A025759 A025760 A025761

Keyword

nonn

Author

Olivier Gérard

Status

approved