A114242 a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)(2n+5)/720.

1, 14, 90, 385, 1274, 3528, 8568, 18810, 38115, 72358, 130130, 223587, 369460, 590240, 915552, 1383732, 2043621, 2956590, 4198810, 5863781, 8065134, 10939720, 14651000, 19392750, 25393095, 32918886, 42280434, 53836615, 68000360

Offset: 0

Author

Emeric Deutsch, Nov 18 2005

Keywords

• nonn
• easy

Comments

Kekulé numbers for certain benzenoids.

Partial sums of A114244. First differences of A006857. - Peter Bala, Sep 21 2007

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/2 and p. 105, eq. (ii) K(Ob(2,4,n))).

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A005585, A006542, A107891.

Programs

Maple

a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n+5)/720: seq(a(n),n=0..30);

Mathematica

Table[((n+1)(n+2)^2 (n+3)^2 (n+4)(2n+5))/720,{n,0,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,14,90,385,1274,3528,8568,18810},30] (* Harvey P. Dale, Aug 21 2013 *)

PARI

a(n)=(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n+5)/720 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1+x)(1 + 5x + x^2)/(1-x)^8.

a(n-2) = (1/6) * Sum_{1 <= x_1, x_2 <= n} (x_1)^2*x_2*(det V(x_1,x_2))^2 = (1/6)*Sum_{1 <= i,j <= n} i^2*j*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Aug 21 2013

From Amiram Eldar, May 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 3550 - 5120*log(2).

Sum_{n>=0} (-1)^n/a(n) = 3430 - 1280*Pi + 60*Pi^2. (End)