OFFSET
0,2
COMMENTS
Kekulé numbers for certain benzenoids.
a(n-4), n>=4, is the number of ways to have n identical objects in m=4 of altogether n distinguishable boxes (n-4 boxes stay empty). - Wolfdieter Lang, Nov 13 2007
REFERENCES
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 23).
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
FORMULA
a(n) = C(n+4,4)*C(n+3,2)(n+1)/3. - Paul Barry, May 13 2006
G.f.: (1+12*x+18*x^2+4*x^3)/(1-x)^8.
a(n) = 4*C(n,4)^2/n, n >= 4. - Zerinvary Lajos, May 09 2008
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 20*Pi^2 - 589/3.
Sum_{n>=0} (-1)^n/a(n) = 64*log(2) - 2*Pi^2 - 71/3. (End)
E.g.f.: (144 + 2736*x + 7992*x^2 + 7416*x^3 + 2826*x^4 + 486*x^5 + 37*x^6 + x^7)*exp(x)/144. - G. C. Greubel, Oct 28 2022
EXAMPLE
a(2) = 150 because n=6 identical balls can be put into m=4 of n=6 distinguishable boxes in binomial(6,4)*(4!/(3!*1!)+ 4!/(2!*2!)) = 15*(4 + 6) = 150 ways. The m=4 part partitions of 6, namely (1^3,3) and (1^2,2^2) specify the filling of each of the 15 possible four box choices. - Wolfdieter Lang, Nov 13 2007
MAPLE
a:=(n+1)^2*(n+2)^2*(n+3)^2*(n+4)/144: seq(a(n), n=0..30);
MATHEMATICA
Array[Binomial[# + 4, 4] Binomial[# + 3, 2] (# + 1)/3 &, 28, 0] (* or *)
CoefficientList[Series[(1 + 12 x + 18 x^2 + 4 x^3)/(1 - x)^8, {x, 0, 27}], x] (* Michael De Vlieger, Dec 17 2017 *)
PROG
(MuPAD) 4*binomial(n, 4)^2/n $ n = 4..35; // Zerinvary Lajos, May 09 2008
(Haskell)
a108647 = flip a103371 3 . (+ 3) -- Reinhard Zumkeller, Apr 04 2014
(Magma) [4*Binomial(n+4, 4)^2/(n+4): n in [0..30]]; // G. C. Greubel, Oct 28 2022
(SageMath) [4*binomial(n+4, 4)^2/(n+4) for n in (0..30)] # G. C. Greubel, Oct 28 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 13 2005
STATUS
approved