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%I #56 Sep 04 2022 04:05:04
%S 0,16,64,160,320,560,896,1344,1920,2640,3520,4576,5824,7280,8960,
%T 10880,13056,15504,18240,21280,24640,28336,32384,36800,41600,46800,
%U 52416,58464,64960,71920,79360,87296,95744,104720,114240,124320,134976,146224
%N Number of ways to change three non-identical letters in the word aabbccdd..., where there are n types of letters.
%C There are two ways to change abc: abc -> bca and abc -> cab, that's why we get 2*C(2n,3). There are 2n*(2n-2) = 4n*(n-1) = 8*C(n,2) cases when the two chosen letters are identical, that's why we get -8*C(n,2). Thanks to _Miklos Kristof_ for help.
%C A diagonal of A059056. - _Zerinvary Lajos_, Jun 18 2007
%C With offset "1", a(n) is 16 times the self convolution of n. - _Wesley Ivan Hurt_, Apr 06 2015
%C Number of orbits of Aut(Z^7) as function of the infinity norm (n+2) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 53760. - _Philippe A.J.G. Chevalier_, Dec 28 2015
%H Stefano Spezia, <a href="/A102860/b102860.txt">Table of n, a(n) for n = 2..10000</a>
%H Mark Roger Sepanski, <a href="https://doi.org/10.37236/3350">On Divisibility of Convolutions of Central Binomial Coefficients</a>, Electronic Journal of Combinatorics, 21 (1) 2014, Article P1.32.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 16*C(n, 3) = 2*C(2*n, 3) - 8*C(n, 2).
%F From _R. J. Mathar_, Mar 09 2009: (Start)
%F G.f.: 16*x^3/(1-x)^4.
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
%F a(n) = 8*n*(n-1)*(n-2)/3. (End)
%F a(n) = 16*A000292(n-2). - _J. M. Bergot_, May 29 2014
%F E.g.f.: 8*exp(x)*x^3/3. - _Stefano Spezia_, May 19 2021
%F From _Amiram Eldar_, Sep 04 2022: (Start)
%F Sum_{n>=3} 1/a(n) = 3/32.
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 3*(8*log(2)-5)/32. (End)
%e a(4) = 64 = 2*C(8,3) - 8*C(4,2) = 2*56 - 8*6 = 112 - 48.
%p A102860:=n->8*n*(n-1)*(n-2)/3: seq(A102860(n), n=2..50); # _Wesley Ivan Hurt_, Apr 06 2015
%t Table[8n(n-1)(n-2)/3,{n,2,50}] (* _Wesley Ivan Hurt_, Apr 06 2015 *)
%t LinearRecurrence[{4,-6,4,-1},{0,16,64,160},50] (* _Harvey P. Dale_, May 20 2021 *)
%o (Magma) [8*n*(n-1)*(n-2)/3 : n in [2..50]]; // _Wesley Ivan Hurt_, Apr 06 2015
%o (PARI) concat([0],Vec(16*x^3/(1-x)^4+O(x^40))) \\ _Stefano Spezia_, May 22 2021
%Y Cf. A000292, A046092, A059056.
%K easy,nonn
%O 2,2
%A _Zerinvary Lajos_, Mar 01 2005
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