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 A102860 Number of ways to change three non-identical letters in the word aabbccdd..., where there are n types of letters. 13
 0, 16, 64, 160, 320, 560, 896, 1344, 1920, 2640, 3520, 4576, 5824, 7280, 8960, 10880, 13056, 15504, 18240, 21280, 24640, 28336, 32384, 36800, 41600, 46800, 52416, 58464, 64960, 71920, 79360, 87296, 95744, 104720, 114240, 124320, 134976, 146224 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS 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. A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007 With offset "1", a(n) is 16 times the self convolution of n. - Wesley Ivan Hurt, Apr 06 2015 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 LINKS Stefano Spezia, Table of n, a(n) for n = 2..10000 Mark Roger Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, Article P1.32. Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA a(n) = 16*C(n, 3) = 2*C(2*n, 3) - 8*C(n, 2). From R. J. Mathar, Mar 09 2009: (Start) G.f.: 16*x^3/(1-x)^4. a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). a(n) = 8*n*(n-1)*(n-2)/3. (End) a(n) = 16*A000292(n-2). - J. M. Bergot, May 29 2014 E.g.f.: 8*exp(x)*x^3/3. - Stefano Spezia, May 19 2021 From Amiram Eldar, Sep 04 2022: (Start) Sum_{n>=3} 1/a(n) = 3/32. Sum_{n>=3} (-1)^(n+1)/a(n) = 3*(8*log(2)-5)/32. (End) EXAMPLE a(4) = 64 = 2*C(8,3) - 8*C(4,2) = 2*56 - 8*6 = 112 - 48. MAPLE A102860:=n->8*n*(n-1)*(n-2)/3: seq(A102860(n), n=2..50); # Wesley Ivan Hurt, Apr 06 2015 MATHEMATICA Table[8n(n-1)(n-2)/3, {n, 2, 50}] (* Wesley Ivan Hurt, Apr 06 2015 *) LinearRecurrence[{4, -6, 4, -1}, {0, 16, 64, 160}, 50] (* Harvey P. Dale, May 20 2021 *) PROG (Magma) [8*n*(n-1)*(n-2)/3 : n in [2..50]]; // Wesley Ivan Hurt, Apr 06 2015 (PARI) concat([0], Vec(16*x^3/(1-x)^4+O(x^40))) \\ Stefano Spezia, May 22 2021 CROSSREFS Cf. A000292, A046092, A059056. Sequence in context: A016802 A309573 A205064 * A136264 A266103 A100184 Adjacent sequences: A102857 A102858 A102859 * A102861 A102862 A102863 KEYWORD easy,nonn AUTHOR Zerinvary Lajos, Mar 01 2005 STATUS approved

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Last modified December 3 08:47 EST 2023. Contains 367531 sequences. (Running on oeis4.)