OFFSET
0,3
COMMENTS
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 19 2005
a(n-2), n>=3, is the number of ways to have n identical objects in m=3 of altogether n distinguishable boxes (n-3 boxes stay empty). - Wolfdieter Lang, Nov 13 2007
Starting with offset 1 = row sums of triangle A096948 and binomial transform of [1, 11, 37, 55, 38, 10, 0, 0, 0, ...]. - Gary W. Adamson, Aug 08 2008
REFERENCES
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 233, # 11).
T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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]."]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
From Paul Barry, Feb 03 2005: (Start)
G.f.: x*(1 + 6*x + 3*x^2)/(1 - x)^6.
a(n) = C(n, 2)*C(n+1, 3). (End)
a(n) = 3*C(n+2,3)^2/(n+2). - Zerinvary Lajos, May 09 2008
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. - Harvey P. Dale, Oct 19 2014
a(n) = Sum_{k=0..n} Sum_{i=0..n} i*C(k+1,k-1). - Wesley Ivan Hurt, Sep 21 2017
a(n) = Sum_{i=0..n} (n+2)*(n-i)^3/3. - Bruno Berselli, Oct 31 2017
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*Pi^2 - 57/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 45/2 - Pi^2/2 - 24*log(2). (End)
E.g.f.: exp(x)*x*(12 + 60*x + 54*x^2 + 14*x^3 + x^4)/12. - Stefano Spezia, May 22 2023
EXAMPLE
a(3)=60 because n=5 identical balls can be put into m=3 of n=5 distinguishable boxes in binomial(5,3)*(3!/(2!*1!)+ 3!/(1!*2!) ) = 10*(3+3) = 60 ways. The m=3 part partitions of 5, namely (1^2,3) and (1,2^2) specify the filling of each of the 10 possible three-box choices. - Wolfdieter Lang, Nov 13 2007
MAPLE
a:=n->n^2*(n+1)^2*(n+2)/12: seq(a(n), n=0..33); # Emeric Deutsch, Jun 19 2005
MATHEMATICA
Table[n^2 (n+1)^2 (n+2)/12, {n, 0, 30}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 12, 60, 200, 525}, 30] (* Harvey P. Dale, Oct 19 2014 *)
PROG
(Magma) [n^2*(n+1)^2*(n+2)/12: n in [0..40]]; // Vincenzo Librandi, May 22 2011
(Haskell)
a004302 0 = 0
a004302 n = a103371 (n + 1) 2 -- Reinhard Zumkeller, Apr 04 2014
(PARI) a(n)=n^2*(n+1)^2*(n+2)/12 \\ Charles R Greathouse IV, Oct 07 2015
(SageMath)
def A004302(n): return 3*binomial(n+2, 3)^2//(n+2)
print([A004302(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved