0,3

Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 19 2005

a(n-2), n>=3, is the number of ways to have n identical objects in m=3 of alltogether n distinguishable boxes (n-3 boxes stay empty). W. 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,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 08 2008]

A004302(n)=Product of sum of first n Triangular numbers and Triangular number(n). [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.233, # 11).

a(n)=C(n, 2)C(n+1, 3). G.f.: x(1+6x+3x^2)/(1-x)^6 - Paul Barry (pbarry(AT)wit.ie), Feb 03 2005

a(n)=3*C(n,3)^2/n, n>= 2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008

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. W. Lang, Nov 13 2007.

a:=n->(n+1)^2*(n+2)^2*(n+3)/12: seq(a(n), n=-1..33); (Deutsch)

[seq (stirling2(n+1, n)*binomial(n+2, 3), n=0..29)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2006

a:=n->sum(numbperm (n, 2)*numbcomb (n, 2)/6, j=0..n): seq(a(n), n=1..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

a:=n->1/2*sum(sum (binomial(n, 3)+binomial(n, 2), j=2..n), k=1..n): seq(a(n), n=1..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007

Clear[lst, n, a, f]; f[n_]:=n*(n+1)/2; a=0; lst={}; Do[a+=f[n]; AppendTo[lst, a*f[n]], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]

(Mupad) 3*binomial(n, 3)^2/n $ n = 2..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008

Third column of triangle A103371.

A096948 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 08 2008]

Sequence in context: A033486 A112415 A061624 this_sequence A000554 A012289 A012583

Adjacent sequences: A004299 A004300 A004301 this_sequence A004303 A004304 A004305

nonn

N. J. A. Sloane (njas(AT)research.att.com).