OFFSET
0,2
COMMENTS
Number of permutations of n+5 that avoid the pattern 132 and have exactly 4 descents.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Partial sums of A114242. - Peter Bala, Sep 21 2007
Dimensions of certain Lie algebra (see reference for precise definition).
REFERENCES
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/1).
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 239.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.3, case a=4]
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
From - Vladeta Jovovic, Jan 29 2003: (Start)
a(n) = (4+n)!*(5+n)!/(2880*n!*(n+1)!).
E.g.f.: 1/2880*(2880 + 40320*x + 109440*x^2 + 105120*x^3 + 45000*x^4 + 9504*x^5 + 1016*x^6 + 52*x^7 + x^8)*exp(x). (End)
From Mike Zabrocki, Aug 26 2004: (Start)
a(n) = C(n+5,8) + 6*C(n+6,8) + 6*C(n+7,8) + C(n+8,8).
a(n) = C(n+4,4)*C(n+5,4)/5.
O.g.f.: (1 + 6*x + 6*x^2 + x^3)/(1-x)^9. (End)
From Wolfdieter Lang, Nov 13 2007: (Start)
a(n) = A001263(n+5,5).
Numerator polynomial of the g.f is the fourth row polynomial of the Narayana triangle. (End)
a(n)= C(n+4,4)^2 - C(n+4,3)*C(n+4,5). - Gary Detlefs, Dec 05 2011
From Amiram Eldar, Oct 19 2020: (Start)
Sum_{n>=0} 1/a(n) = 25 * (79 - 8*Pi^2).
Sum_{n>=0} (-1)^n/a(n) = 595/3 - 20*Pi^2. (End)
MAPLE
a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)/2880: seq(a(n), n=0..38); # Emeric Deutsch, Nov 18 2005
MATHEMATICA
Table[Binomial[n+5, 5] * Binomial[n+5, 4]/(n+5), {n, 0, 50}] (* T. D. Noe, May 29 2012 *)
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 15, 105, 490, 1764, 5292, 13860, 32670, 70785}, 40] (* Harvey P. Dale, Oct 19 2024 *)
PROG
(PARI) a(n) = binomial(n+5, 5) * binomial(n+5, 4)/(n+5) \\ Charles R Greathouse IV, Jun 11 2015
(PARI) Vec((1+6*x+6*x^2+x^3)/(1-x)^9 + O(x^99)) \\ Altug Alkan, Sep 01 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Jan 29 2003
Better description from Mike Zabrocki, Aug 26 2004
New definition from N. J. A. Sloane, Aug 28 2010
Zabrocki formulas offset corrected by Gary Detlefs, Dec 05 2011
STATUS
approved