OFFSET
5,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 5..1000
R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
FORMULA
G.f.: z^5*M^6, where M=1+z*M+z^2*M^2 is the g.f. for the Motzkin numbers (A001006). - Emeric Deutsch, Aug 13 2004
a(n) = (sqrt(-3)/81)*((-1)^n*n*(4*n^3-15*n^2-55*n+102)/(n+7)/(n+3)/(n+2)*hypergeom([1/2, n+7],[3],4/3)-(-1)^n*(4*n^4-17*n^3+23*n^2+ 242*n-288)/(n+7)/(n+3)/(n+2)*hypergeom([1/2, n+6],[3],4/3)). - Mark van Hoeij, Oct 29 2011.
a(n) (n + 11) (n - 1) = (n + 4) (3 n + 9) a(n - 2) + (n + 4) (2 n + 9) a(n - 1). - Simon Plouffe, Feb 09 2012
a(n) ~ 3^(n+5/2)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Oct 05 2012
a(n) = 6*sum(j=ceiling((n-5)/2)..(n+1), C(j,2*j-n+5)*C(n+1,j))/(n+1). - Vladimir Kruchinin, Mar 17 2014
MATHEMATICA
RecurrenceTable[{3(-1+n)*n*a[-2+n]+n*(1+2n)*a[-1+n]-(-5+n)*(7+n)*a[n]==0, a[5]==1, a[6]==6}, a, {n, 5, 20}] (* Vaclav Kotesovec, Oct 05 2012 *)
a = DifferenceRoot[Function[{b, n}, {(-2n^2 - 25n - 78)b[n+1] - 3(n+5)(n+6) b[n] + (n+1)(n+13)b[n+2] == 0, b[1] == 1, b[2] == 6}]][# - 4]&;
Table[a[n], {n, 5, 31}] (* Jean-François Alcover, Jan 24 2019 *)
PROG
(Maxima)
a(n) := 6*sum(binomial(j, 2*j-n+5)*binomial(n+1, j), j, ceiling((n-5)/2), (n+1))/(n+1);
/* Vladimir Kruchinin, Mar 18 2014 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vincenzo Librandi, May 03 2013
STATUS
approved