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

Author

Eric W. Weisstein

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[Binomial(n, 5)*5^(n-5): n in [5..25]]; // Vincenzo Librandi, Aug 12 2017

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 *)

PARI

a(n)=binomial(n, 5)*5^(n-5) \\ Charles R Greathouse IV, Sep 03 2011

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)