A050982 5-idempotent numbers.

1, 30, 525, 7000, 78750, 787500, 7218750, 61875000, 502734375, 3910156250, 29326171875, 213281250000, 1510742187500, 10458984375000, 70971679687500, 473144531250000, 3105010986328125, 20091247558593750, 128360748291015625, 810699462890625000

Offset

5,2

Comments

Number of n-permutations of 6 objects: t,u,v,z,x, y with repetition allowed, containing exactly five u's. Example: a(6)=30 because we have uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu, uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu, uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu, uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu, uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu. - Zerinvary Lajos, Jun 16 2008

References

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43.

Links

Vincenzo Librandi, Table of n, a(n) for n = 5..1000

Eric Weisstein's World of Mathematics, Idempotent Number.

Index entries for linear recurrences with constant coefficients, signature (30,-375,2500,-9375,18750,-15625).

Formula

a(n) = C(n, 5)*5^(n-5).

G.f.: x^5/(1-5*x)^6. - Zerinvary Lajos, Aug 06 2008

From Amiram Eldar, Apr 17 2022: (Start)

Sum_{n>=5} 1/a(n) = 6400*log(5/4) - 17125/12.

Sum_{n>=5} (-1)^(n+1)/a(n) = 32400*log(6/5) - 23625/4. (End)

Maple

seq(binomial(n, 5)*5^(n-5), n=5..32); # Zerinvary Lajos, Jun 16 2008

Mathematica

CoefficientList[Series[1 / (1 - 5 x)^6, {x, 0, 33}], x] (* Vincenzo Librandi, Aug 12 2017 *)

Prog

(PARI) a(n)=binomial(n, 5)*5^(n-5) \\ Charles R Greathouse IV, Sep 03 2011
(Magma) [Binomial(n, 5)*5^(n-5): n in [5..25]]; // Vincenzo Librandi, Aug 12 2017

Crossrefs

Cf. A001788, A036216, A040075, A050988, A050989, A000389, A054849, A036219, A045543, A036084, A140404, A000389, A054849, A036219, A045543, A036084, A140404.

Sequence in context: A020926 A229563 A022754 * A004327 A270499 A286975

Adjacent sequences: A050979 A050980 A050981 * A050983 A050984 A050985

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved