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

%I A004302 #90 Mar 12 2025 08:06:41
%S A004302 0,1,12,60,200,525,1176,2352,4320,7425,12100,18876,28392,41405,58800,
%T A004302 81600,110976,148257,194940,252700,323400,409101,512072,634800,780000,
%U A004302 950625,1149876,1381212,1648360,1955325,2306400,2706176,3159552,3671745,4248300,4895100
%N A004302 a(n) = n^2*(n+1)^2*(n+2)/12.
%C A004302 Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 19 2005
%C A004302 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
%C A004302 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
%D A004302 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 233, # 11).
%D A004302 T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
%H A004302 Vincenzo Librandi, <a href="/A004302/b004302.txt">Table of n, a(n) for n = 0..1000</a>
%H A004302 Paolo Aluffi, <a href="https://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, 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]."]
%H A004302 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A004302 From _Paul Barry_, Feb 03 2005: (Start)
%F A004302 G.f.: x*(1 + 6*x + 3*x^2)/(1 - x)^6.
%F A004302 a(n) = C(n, 2)*C(n+1, 3). (End)
%F A004302 a(n) = 3*C(n+2,3)^2/(n+2). - _Zerinvary Lajos_, May 09 2008
%F A004302 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
%F A004302 a(n) = A000217(n)*A000292(n). - _Bruno Berselli_, Jan 13 2015
%F A004302 a(n) = Sum_{k=0..n} Sum_{i=0..n} i*C(k+1,k-1). - _Wesley Ivan Hurt_, Sep 21 2017
%F A004302 a(n) = Sum_{i=0..n} (n+2)*(n-i)^3/3. - _Bruno Berselli_, Oct 31 2017
%F A004302 From _Amiram Eldar_, May 29 2022: (Start)
%F A004302 Sum_{n>=1} 1/a(n) = 3*Pi^2 - 57/2.
%F A004302 Sum_{n>=1} (-1)^(n+1)/a(n) = 45/2 - Pi^2/2 - 24*log(2). (End)
%F A004302 E.g.f.: exp(x)*x*(12 + 60*x + 54*x^2 + 14*x^3 + x^4)/12. - _Stefano Spezia_, May 22 2023
%e A004302 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
%p A004302 a:=n->n^2*(n+1)^2*(n+2)/12: seq(a(n),n=0..33); # _Emeric Deutsch_, Jun 19 2005
%t A004302 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 *)
%o A004302 (Magma) [n^2*(n+1)^2*(n+2)/12: n in [0..40]]; // _Vincenzo Librandi_, May 22 2011
%o A004302 (Haskell)
%o A004302 a004302 0 = 0
%o A004302 a004302 n = a103371 (n + 1) 2 -- _Reinhard Zumkeller_, Apr 04 2014
%o A004302 (PARI) a(n)=n^2*(n+1)^2*(n+2)/12 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A004302 (SageMath)
%o A004302 def A004302(n): return 3*binomial(n+2,3)^2//(n+2)
%o A004302 print([A004302(n) for n in range(41)]) # _G. C. Greubel_, Mar 12 2025
%Y A004302 Third column of triangle A103371.
%Y A004302 Main diagonal of A103252.
%Y A004302 Cf. A000217, A000292, A096948.
%K A004302 nonn,easy
%O A004302 0,3
%A A004302 _N. J. A. Sloane_