A004302 a(n) = n^2*(n+1)^2*(n+2)/12.

0, 1, 12, 60, 200, 525, 1176, 2352, 4320, 7425, 12100, 18876, 28392, 41405, 58800, 81600, 110976, 148257, 194940, 252700, 323400, 409101, 512072, 634800, 780000, 950625, 1149876, 1381212, 1648360, 1955325, 2306400, 2706176, 3159552, 3671745, 4248300, 4895100

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

Third column of triangle A103371.

Main diagonal of A103252.

Cf. A000217, A000292, A096948.

Programs

Haskell

a004302 0 = 0
a004302 n = a103371 (n + 1) 2 -- Reinhard Zumkeller, Apr 04 2014

Magma

[n^2*(n+1)^2*(n+2)/12: n in [0..40]]; // Vincenzo Librandi, May 22 2011

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 *)

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

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) = A000217(n)*A000292(n). - Bruno Berselli, Jan 13 2015

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