|
| |
|
|
A006542
|
|
C(n,3)*C(n-1,3)/4.
(Formerly M4707)
|
|
17
| |
|
|
1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575, 2992626, 3708810, 4562425, 5573800, 6765440, 8162176
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 4,2
|
|
|
COMMENTS
| Number of permutations of n+4 which avoid the pattern 132 and have exactly 3 descents. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 20 2005
a(n)=number of Dyck n-paths with exactly 4 peaks. - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006
Six-dimensional figurate numbers for a hyperpyramid with pentagonal base. This corresponds to the sum(sum(sum(sum(1+sum(5*n))))) interpretation, see the Munafo webpage. [From Robert Munafo (mrob27(AT)gmail.com), Jun 18 2009]
|
|
|
REFERENCES
| S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.166, no.1).
G. Kreweras, Traitemant simultane du "Probleme de Young" et du "Probleme de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle. Institut de Statistique, Universit\'{e} de Paris, 10 (1967), 23-31.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| R. Munafo, C(n,3)C(n-1,3)/4 [From Robert Munafo (mrob27(AT)gmail.com), Jun 18 2009]
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
|
|
|
FORMULA
| C(n, 3)C(n-1, 3)/4 = n ((n-1)(n-2))^2 (n-3)/144.
E.g.f.: 1/144*x^4*(6+6*x+x^2)*exp(x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 29 2003
a(n) = sum(sum(sum(sum(1 + sum(5*n))))) = sum (A006414) - Xavier Acloque Oct 08 2003
a(n) = C(n, 6) + 3 C(n+1, 6) + C(n+2, 6) o.g.f. (1+3x+x^2)/(1-x)^7 - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
G.f.=z^4*(1+3z+z^2)/(1-z)^7. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 20 2005
C(n-2, n-4)*C(n-1, n-3)*C(n, n-2)/18 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 29 2005
a(n) = C(n,4)C(n,3)/n - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006
a(n+2) = 1/4*sum {1 <= x_1, x_2 <= n} x_1*x_2*(det V(x_1,x_2))^2 = 1/4*sum {1 <= i,j <= n} i*j*(i-j)^2, where V(x_1,x_2} is the Vandermonde matrix of order 2. - Peter Bala (pbala(AT)toucansurf.com), Sep 21 2007
a(n)= C(n-1,3)^2 - C(n-1,2)*C(n-1,4). [From Gary Detlefs, Dec 05 2011]
|
|
|
MAPLE
| A006542:=-(1+3*z+z**2)/(z-1)**7; [Conjectured by S. Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
| a[n_] := Binomial[n, 3]Binomial[n-1, 3]/4
|
|
|
PROG
| (PARI) a(n)=n*((n-1)*(n-2))^2*(n-3)/144
|
|
|
CROSSREFS
| The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288m A134289, A134290, A134291, A140925, A140935, A169937.
Cf. A001263, A005891, A006322, A004068, A006414.
Fourth column of the table of Narayana numbers A001263.
Cf. A005585, A047819, A107891, A114242.
Sequence in context: A008531 A051230 A008413 * A086462 A201830 A192019
Adjacent sequences: A006539 A006540 A006541 * A006543 A006544 A006545
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Zabroki and Lajos formulas offset corrected by Gary Detlefs Dec 05 2011
|
| |
|
|
