0,2

Number of tilings of a <3,n,3> hexagon.

Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 18 2005

Partial sums of A107891. - Peter Bala (pbala(AT)toucansurf.com), Sep 21 2007

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.

J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=2]

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.232, # 2 and p. 105, eq.(ii), K(0a(2,5,n))).

H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects

G.f.: (1+10x+20x^2+10x^3+x^4)/(1-x)^10.

a(n)=C(n,n-1)*C(n+1,n-2)*C(n+2,n-3)/12, n>=3 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2007

a(n-3) = 1/24*sum {1 <= x_1, x_2, x_3 <= n} (det V(x_1,x_2,x_3))^2 = 1/24*sum {1 <= i,j,k <= n} ((i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. - Peter Bala (pbala(AT)toucansurf.com), Sep 21 2007

a:=n->(n+1)*(n+2)^2*(n+3)^3*(n+4)^2*(n+5)/8640: seq(a(n), n=0..30); (Deutsch)

seq(binomial(n, n-1)*binomial(n+1, n-2)*binomial(n+2, n-3)/12, n=3..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2007

(PARI) a(n)=if(n<0, 0, binomial(n+5, 5)*binomial(n+4, 3)*(n+3)/12)

(PARI) a(n)=if(n<0, 0, -1/8*polcoeff(charpoly(matrix(n+3, n+3, i, j, (i-j)^2)), n))

Third row of array A103905.

Cf. A002415, A107891, A107915, A133708.

Sequence in context: A022712 A056128 A027791 this_sequence A163689 A140044 A027332

Adjacent sequences: A047816 A047817 A047818 this_sequence A047820 A047821 A047822

nonn

N. J. A. Sloane (njas(AT)research.att.com).