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A004302
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n^2*(n+1)^2*(n+2)/12.
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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]
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REFERENCES
| 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).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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FORMULA
| a(n)=C(n, 2)*C(n+1, 3). G.f.: x*(1+6*x+3*x^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
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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. W. Lang, Nov 13 2007.
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MAPLE
| a:=n->n^2*(n+1)^2*(n+2)/12: seq(a(n), n=0..33); (Deutsch)
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MATHEMATICA
| 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]
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PROG
| (MAGMA) [n^2*(n+1)^2*(n+2)/12: n in [0..40]]; // Vincenzo Librandi, May 22 2011
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CROSSREFS
| Third column of triangle A103371.
A096948 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 08 2008]
Sequence in context: A112415 A174642 A061624 * A000554 A012289 A012583
Adjacent sequences: A004299 A004300 A004301 * A004303 A004304 A004305
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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