A108647 a(n) = (n+1)^2*(n+2)^2*(n+3)^2*(n+4)/144.

1, 20, 150, 700, 2450, 7056, 17640, 39600, 81675, 157300, 286286, 496860, 828100, 1332800, 2080800, 3162816, 4694805, 6822900, 9728950, 13636700, 18818646, 25603600, 34385000, 45630000, 59889375, 77808276, 100137870, 127747900

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

Fourth column of triangle A103371.

Sequence in context: A100190 A189494 A022680 * A164605 A000492 A015866

Adjacent sequences: A108644 A108645 A108646 * A108648 A108649 A108650

Keyword

nonn

Author

Emeric Deutsch, Jun 13 2005

Status

approved