login
A107908
a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3*n+5)/720.
1
1, 16, 110, 490, 1666, 4704, 11592, 25740, 52635, 100672, 182182, 314678, 522340, 837760, 1303968, 1976760, 2927349, 4245360, 6042190, 8454754, 11649638, 15827680, 21229000, 28138500, 36891855, 47882016, 61566246, 78473710
OFFSET
0,2
COMMENTS
Kekulé numbers for certain benzenoids.
LINKS
Paolo Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 167).
FORMULA
a(0)=1, a(1)=16, a(2)=110, a(3)=490, a(4)=1666, a(5)=4704, a(6)=11592, a(7)=25740, 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 07 2013
G.f.: (1 + 8*x + 10*x^2 + 2*x^3) / (1 - x)^8. - Colin Barker, Apr 22 2020
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=0} 1/a(n) = 3645*sqrt(3)*Pi/28 + 75*Pi^2 - 32805*log(3)/28 - 2245/14.
Sum_{n>=0} (-1)^n/a(n) = 3645*sqrt(3)*Pi/14 - 45*Pi^2/2 - 7680*log(2)/7 - 6065/14. (End)
MAPLE
a:=n->(1/720)*(n+1)*(n+2)^2*(n+3)^2*(n+4)*(3*n+5): seq(a(n), n=0..33);
MATHEMATICA
Table[((n+1)(n+2)^2 (n+3)^2 (n+4)(3n+5))/720, {n, 0, 30}] (* or *) LinearRecurrence[ {8, -28, 56, -70, 56, -28, 8, -1}, {1, 16, 110, 490, 1666, 4704, 11592, 25740}, 30] (* Harvey P. Dale, Aug 07 2013 *)
PROG
(Python)
from itertools import islice
def A107908_generator():
m = [21, -13, 3]+[1]*5
yield m[-1]
while True:
for i in range(7):
m[i+1]+= m[i]
yield m[-1]
list(islice(A107908_generator(), 0, 50, 1)) # Chai Wah Wu, Nov 14 2014
(PARI) Vec((1 + 8*x + 10*x^2 + 2*x^3) / (1 - x)^8 + O(x^30)) \\ Colin Barker, Apr 22 2020
CROSSREFS
Sequence in context: A155871 A120668 A053526 * A177046 A234250 A240786
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 12 2005
STATUS
approved