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A036222
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Expansion of 1/(1-3*x)^9; 9-fold convolution of A000244 (powers of 3).
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12
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1, 27, 405, 4455, 40095, 312741, 2189187, 14073345, 84440070, 478493730, 2583866142, 13389124554, 66945622770, 324428787270, 1529449997130, 7035469986798, 31659614940591, 139674771796725, 605257344452475
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OFFSET
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0,2
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COMMENTS
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With a different offset, number of n-permutations (n>=8) of 4 objects: u, v, z, x with repetition allowed, containing exactly eight (8) u's. Example: a(1)=27 because we have uuuuuuuuv, uuuuuuuuz, uuuuuuuux, uuuuuuuvu, uuuuuuuzu, uuuuuuuxu, uuuuuuvuu, uuuuuuzuu, uuuuuuxuu, uuuuuvuuu, uuuuuzuuu, uuuuuxuuu, uuuuvuuuu, uuuuzuuuu, uuuuxuuuu, uuuvuuuuu, uuuzuuuuu, uuuxuuuuu, uuvuuuuuu, uuzuuuuuu, uuxuuuuuu, uvuuuuuuu, uzuuuuuuu, uxuuuuuuu, vuuuuuuuu, zuuuuuuuu, xuuuuuuuu. - Zerinvary Lajos, Jun 23 2008
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LINKS
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FORMULA
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a(n) = 3^n*binomial(n+8, 8).
G.f.: 1/(1-3*x)^9.
a(0)=1, a(1)=27, a(2)=405, a(3)=4455, a(4)=40095, a(5)=312741, a(6)=2189187, a(7)=14073345, a(8)=84440070, a(n) = 27*a(n-1) - 324*a(n-2) + 2268*a(n-3) - 10206*a(n-4) + 30618*a(n-5) - 61236*a(n-6) + 78732*a(n-7) - 59049*a(n-8) + 19683*a(n-9). - Harvey P. Dale, Jan 07 2016
Sum_{n>=0} 1/a(n) = 43632/35 - 3072*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 393216*log(4/3) - 3959208/35. (End)
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MAPLE
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MATHEMATICA
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CoefficientList[Series[1/(1-3x)^9, {x, 0, 30}], x] (* or *) LinearRecurrence[{27, -324, 2268, -10206, 30618, -61236, 78732, -59049, 19683}, {1, 27, 405, 4455, 40095, 312741, 2189187, 14073345, 84440070}, 30] (* Harvey P. Dale, Jan 07 2016 *)
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PROG
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(Sage) [3^n*binomial(n+8, 8) for n in range(30)] # Zerinvary Lajos, Mar 13 2009
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CROSSREFS
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Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), this sequence (m=8), A036223 (m=9), A172362 (m=10).
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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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