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COMMENTS
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With a different offset, number of n-permutations (n>=6) of 9 objects:p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly six (6) u's.
If n=6 then a(0)= 1
Example: a(1)=56 because we have
uuuuuup, uuuuupu, uuuupuu, uuupuuu, uupuuuu, upuuuuu, puuuuuu,
uuuuuur, uuuuuru, uuuuruu, uuuruuu, uuruuuu, uruuuuu, ruuuuuu,
uuuuuus, uuuuusu, uuuusuu, uuusuuu, uusuuuu, usuuuuu, suuuuuu,
uuuuuut, uuuuutu, uuuutuu, uuutuuu, uutuuuu, utuuuuu, tuuuuuu,
uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu,
uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu,
uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu,
uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu.
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MATHEMATICA
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Table[Binomial[n+6, 6]8^n, {n, 0, 20}] (* or *) LinearRecurrence[ {56, -1344, 17920, -143360, 688128, -1835008, 2097152}, {1, 56, 1792, 43008, 860160, 15138816, 242221056}, 20] (* From Harvey P. Dale, Dec 15 2011 *)
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