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A005324 Column of Motzkin triangle A026300.
(Formerly M3902)
3
1, 5, 20, 70, 230, 726, 2235, 6765, 20240, 60060, 177177, 520455, 1524120, 4453320, 12991230, 37854954, 110218905, 320751445, 933149470, 2714401580, 7895719634, 22969224850, 66829893650, 194486929650, 566141346225, 1648500576021 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

COMMENTS

a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0, s(n) = 4. - Clark Kimberling

a(n) = T(n,n-4), where T is the array in A026300.

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=4..29.

R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449 [cs.DM], 2013.

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, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

FORMULA

G.f.: z^4*M^5, where M is g.f. of Motzkin numbers (A001006).

a(n) = (-5*I*(-1)^n*(n^4-6*n^3-43*n^2-24*n+36)*3^(1/2)*hypergeom([1/2, n+2],[1],4/3)+15*I*(-1)^n*(n^4+6*n^3+17*n^2+24*n-12)*3^(1/2)*hypergeom([1/2, n+1],[1],4/3))/(6*(n+3)*(n+2)*(n+4)*(n+5)*(n+6)). - Mark van Hoeij, Oct 29 2011

a(n) (n + 9) (n - 1) = (n + 3) (3 n + 6) a(n - 2) + (n + 3) (2 n + 7) a(n - 1). - Simon Plouffe, Feb 09 2012

a(n) =  5*sum(j=ceiling((n-4)/2)..(n+1), binomial(j,2*j-n+4)*binomial(n+1,j))/(n+1). - Vladimir Kruchinin, Mar 17 2014

MATHEMATICA

T[n_, k_] := Sum[m = 2j+n-k; Binomial[n, m] (Binomial[m, j] - Binomial[m, j-1]), {j, 0, k/2}];

a[n_] := T[n, n-4];

Table[a[n], {n, 4, 30}] (* Jean-François Alcover, Jul 27 2018 *)

PROG

(Maxima) a(n) := 5*sum(binomial(j, 2*j-n+4)*binomial(n+1, j), j, ceiling((n-4)/2), (n+1))/(n+1); /* Vladimir Kruchinin, Mar 18 2014 */

CROSSREFS

Cf. A026300.

A diagonal of triangle A020474.

Sequence in context: A080930 A169792 A000343 * A304011 A243869 A154638

Adjacent sequences:  A005321 A005322 A005323 * A005325 A005326 A005327

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 23 21:35 EDT 2018. Contains 316541 sequences. (Running on oeis4.)