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A102860
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Number of ways to change three non-identical letters in the word aabbccdd..., where there are n types of letters.
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14
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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
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OFFSET
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2,2
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COMMENTS
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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.
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
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LINKS
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FORMULA
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a(n) = 16*C(n, 3) = 2*C(2*n, 3) - 8*C(n, 2).
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)
Sum_{n>=3} 1/a(n) = 3/32.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3*(8*log(2)-5)/32. (End)
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EXAMPLE
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a(4) = 64 = 2*C(8,3) - 8*C(4,2) = 2*56 - 8*6 = 112 - 48.
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MAPLE
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {0, 16, 64, 160}, 50] (* Harvey P. Dale, May 20 2021 *)
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PROG
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(PARI) concat([0], Vec(16*x^3/(1-x)^4+O(x^40))) \\ Stefano Spezia, May 22 2021
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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