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A054880
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a(n) = 3*(9^n - 1)/4.
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5
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0, 6, 60, 546, 4920, 44286, 398580, 3587226, 32285040, 290565366, 2615088300, 23535794706, 211822152360, 1906399371246, 17157594341220, 154418349070986, 1389765141638880, 12507886274749926, 112570976472749340, 1013138788254744066, 9118249094292696600, 82064241848634269406, 738578176637708424660
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OFFSET
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0,2
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COMMENTS
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Number of walks of length 2n+1 along the edges of a (3 dimensional) cube between two opposite vertices.
Urn A initially contains 3 labeled balls while urn B is empty. A ball is randomly selected and switched from one urn to the other. a(n)/3^(2n+1) is the probability that urn A is empty after 2n+1 switches. - Geoffrey Critzer, May 23 2013
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LINKS
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FORMULA
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G.f.: (3/4)/(1 - 9*x) - (3/4)/(1 - x).
sin(x)^3 = Sum_{k>=0} (-1)^(k+1)*a(k)*x^(2k+1)/(2k+1)!. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
a(n) = 10*a(n-1) - 9*a(n-2) with a(0) = 0 and a(1) = 6. - Miquel A. Fiol, Mar 09 2024
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MATHEMATICA
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Table[(2 n + 1)! Coefficient[Series[Sinh[x]^3, {x, 0, 2 n + 1}],
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PROG
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(PARI) vector(30, n, n--; 3*(9^n -1)/4) \\ G. C. Greubel, Jul 14 2019
(Magma) [3*(9^n -1)/4: n in [0..30]]; // G. C. Greubel, Jul 14 2019
(Sage) [3*(9^n -1)/4 for n in (0..30)] # G. C. Greubel, Jul 14 2019
(GAP) List([0..30], n-> 3*(9^n -1)/4) # G. C. Greubel, Jul 14 2019
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CROSSREFS
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KEYWORD
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nonn,easy,walk
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AUTHOR
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Paolo Dominici (pl.dm(AT)libero.it), May 23 2000
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STATUS
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approved
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