4,2

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

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.

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. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.166, no.1).

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.

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.

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.yu), Jan 29 2003

a(n) = sum(sum(sum(sum(1 + sum(5*n))))) = sum (A006414) - Xavier Acloque Oct 08 2003

a(n) = C(n+4, 6) + 3 C(n+5, 6) + C(n+6, 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(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n)/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

A006542:=-(1+3*z+z**2)/(z-1)**7; [Conjectured by S. Plouffe in his 1992 dissertation.]

a[n_] := Binomial[n, 3]Binomial[n-1, 3]/4

(PARI) a(n)=n*((n-1)*(n-2))^2*(n-3)/144

Cf. A001263, A005891, A006322, A004068, A006414.

Fourth column of the table of Narayana numbers A001263.

Cf. A005585, A047819, A107891, A114242.

Adjacent sequences: A006539 A006540 A006541 this_sequence A006543 A006544 A006545

Sequence in context: A008531 A051230 A008413 this_sequence A086462 A003207 A095687

nonn,easy

njas