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A004302
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a(n) = n^2*(n+1)^2*(n+2)/12.
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9
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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, 2306400, 2706176, 3159552, 3671745, 4248300, 4895100
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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]."]
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FORMULA
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G.f.: x*(1 + 6*x + 3*x^2)/(1 - x)^6.
a(n) = C(n, 2)*C(n+1, 3). (End)
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
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
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EXAMPLE
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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
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MAPLE
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a:=n->n^2*(n+1)^2*(n+2)/12: seq(a(n), n=0..33); # Emeric Deutsch, Jun 19 2005
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a004302 0 = 0
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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