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With a different offset, number of n-permutations (n=5) of 8 objects s, t, u, v, w, z, x, y with repetition allowed, containing exactly four (4) u's. Example: a(1)=35 because we have
(Sage)[lucas_number2(n, 7, 0)*binomial(n, 4)/7^4for n in range(4, 21)] # Zerinvary Lajos, Mar 13 2009
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1, 35, 735, 12005, 168070, 2117682, 24706290, 271769190, 2853576495, 28852829005, 282757724249, 2699051004195, 25191142705820, 230595844768660, 2075362602917940, 18401548412539068, 161013548609716845, 1392293626213433895, 11911845468714934435, 100937216866479181265
<a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (35,-490,3430,-12005,16807).
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 2800/3 - 6048*log(7/6).
Sum_{n>=0} (-1)^n/a(n) = 14336*log(8/7) - 5740/3. (End)
Table[7^n * Binomial[n+4, 4], {n, 0, 20}] (* Amiram Eldar, Aug 28 2022 *)
(MAGMAMagma) [7^n* Binomial(n+4, 4): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011
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(Sage)[lucas_number2(n, 7, 0)*binomial(n, 4)/7^4for n in xrangerange(4, 21)] # Zerinvary Lajos, Mar 13 2009
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Binomiala(n) = binomial(n+4, 4)*7^n.
(Sage)[lucas_number2(n, 7, 0)*binomial(n, 4)/7^4for n in xrange(4, 21)] # [From __Zerinvary Lajos_, Mar 13 2009]
(MAGMA) [7^n* Binomial(n+4, 4): n in [0..20]]; // _Vincenzo Librandi, _, Oct 12 2011
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