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A005325 Column of Motzkin triangle.
(Formerly M4176)
4
1, 6, 27, 104, 369, 1242, 4037, 12804, 39897, 122694, 373581, 1128816, 3390582, 10136556, 30192102, 89662216, 265640691, 785509362, 2319218869, 6839057544, 20147488020, 59306494520, 174466248840, 512987904000, 1507780192035, 4430417492826, 13015498076181 (list; graph; refs; listen; history; text; internal format)
OFFSET
5,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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
A diagonal of triangle A020474.
Sequence in context: A169793 A054457 A000395 * A099623 A119852 A220529
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vincenzo Librandi, May 03 2013
STATUS
approved

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Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)