|
| |
|
|
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
|
R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Simon Plouffe, < a href="http://arxiv.org/ftp/arxiv/papers/0912/0912.0072.pdf"> Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
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.
|
|
|
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]
|
|
|
CROSSREFS
|
Cf. A026300.
A diagonal of triangle A020474.
Sequence in context: A080930 A169792 A000343 * A154638 A054889 A056384
Adjacent sequences: A005321 A005322 A005323 * A005325 A005326 A005327
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
