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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

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.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. 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 *)

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

Cf. A026300, A026126, A001006.

A diagonal of triangle A020474.

Sequence in context: A169793 A054457 A000395 * A099623 A119852 A220529

Adjacent sequences:  A005322 A005323 A005324 * A005326 A005327 A005328

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vincenzo Librandi, May 03 2013

STATUS

approved

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Last modified October 20 21:31 EDT 2018. Contains 316404 sequences. (Running on oeis4.)