A050989 -id:A050989 - cp's OEIS frontend

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

Eric W. Weisstein

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

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

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).

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

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

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

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

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

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)

A050988 6-idempotent numbers.

Original entry on oeis.org

1, 42, 1008, 18144, 272160, 3592512, 43110144, 480370176, 5043886848, 50438868480, 484213137408, 4489976365056, 40409787285504, 354362750042112, 3037395000360960, 25514118003032064, 210491473525014528, 1708695490967764992, 13669563927742119936, 107917609955858841600

Offset: 6

Eric W. Weisstein

Number of n-permutations of 7 objects: s, t, u, v, z, x, y with repetition allowed, containing exactly six u's. - Zerinvary Lajos, Jun 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 6..400
Eric Weisstein's World of Mathematics, Idempotent Number.

Cf. A000027, A001788, A036216, A040075, A050982, A050989.

seq(binomial(n, 6)*6^(n-6), n=6..32); # Zerinvary Lajos, Jun 16 2008
A050988:=x^6/(1-6*x)^7; series(A050988,x,32); # Zerinvary Lajos, Aug 09 2008

a[n_] := Binomial[n, 6]*6^(n - 6); Array[a, 20, 6] (* Amiram Eldar, Apr 17 2022 *)

a(n) = binomial(n,6)*6^(n-6).
G.f.: x^6/(1-6*x)^7. - Zerinvary Lajos, Aug 09 2008
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=6} 1/a(n) = 102561/5 - 112500*log(6/5).
Sum_{n>=6} (-1)^n/a(n) = 605052*log(7/6) - 466341/5. (End)

A141054 8-idempotent numbers: a(n) = binomial(n+8,8)*8^n.

Original entry on oeis.org

1, 72, 2880, 84480, 2027520, 42172416, 787218432, 13495173120, 215922769920, 3262832967680, 46984794734592, 649244436332544, 8656592484433920, 111869810568069120, 1406363332855726080, 17251390216363573248, 207016682596362878976, 2435490383486622105600

Offset: 0

Zerinvary Lajos, Aug 01 2008

With a different offset, number of n-permutations of 9 objects:
p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight (8) u's. Example: a(1)=72 because we have
uuuuuuuup, uuuuuuupu, uuuuuupuu, uuuuupuuu, uuuupuuuu, uuupuuuuu, uupuuuuuu, upuuuuuuu, puuuuuuuu,
uuuuuuuur, uuuuuuuru, uuuuuuruu, uuuuuruuu, uuuuruuuu, uuuruuuuu, uuruuuuuu, uruuuuuuu, ruuuuuuuu,
uuuuuuuus, uuuuuuusu, uuuuuusuu, uuuuusuuu, uuuusuuuu, uuusuuuuu, uusuuuuuu, usuuuuuuu, suuuuuuuu,
uuuuuuuut, uuuuuuutu, uuuuuutuu, uuuuutuuu, uuuutuuuu, uuutuuuuu, uutuuuuuu, utuuuuuuu, tuuuuuuuu,
uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu, vuuuuuuuu,
uuuuuuuuz, uuuuuuuzu, uuuuuuzuu, uuuuuzuuu, uuuuzuuuu, uuuzuuuuu, uuzuuuuuu, uzuuuuuuu, zuuuuuuuu,
uuuuuuuux, uuuuuuuxu, uuuuuuxuu, uuuuuxuuu, uuuuxuuuu, uuuxuuuuu, uuxuuuuuu, uxuuuuuuu, xuuuuuuuu,
uuuuuuuuy, uuuuuuuyu, uuuuuuyuu, uuuuuyuuu, uuuuyuuuu, uuuyuuuuu, uuyuuuuuu, uyuuuuuuu, yuuuuuuuu.

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000027, A001788, A036216, A040075, A050982, A050988, A050989, A059297, A059300.

[8^n* Binomial(n+8, 8): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Maple
```
seq(binomial(n+8,8)*8^n, n=0..17);
```

Table[Binomial[n + 8, 8] 8^n, {n, 0, 15}] (* Michael De Vlieger, Jul 24 2017 *)

vector(15,n,binomial(n+7,8)*8^(n-1)) \\ Derek Orr, Jul 24 2017

a(n) = binomial(n+8,8)*8^n.
G.f.: 1/(1-8*x)^9. - Vincenzo Librandi, Oct 16 2011
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=0} 1/a(n) = 738990736/105 - 52706752*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 306110016*log(9/8) - 1261909808/35. (End)

A170932 a(n) = binomial(n + 8, 8)*7^n .

Original entry on oeis.org

1, 63, 2205, 56595, 1188495, 21630609, 353299947, 5299499205, 74192988870, 980996186170, 12360551945742, 149450309889426, 1743586948709970, 19715944727720430, 216875392004924730, 2327795874186192102, 24441856678955017071, 251607348165713411025

Offset: 0

Zerinvary Lajos, Feb 08 2010

With a different offset, number of n-permutations of 8 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight, (8) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (63,-1764,28812,-302526,2117682,-9882516,29647548,-51883209,40353607).

Cf. A027474, A140107, A139641, A140404, A036226, A050989.

[Binomial(n + 8, 8)*7^n: n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 8, 8]*7^n, {n, 0, 20}]

a(n) = C(n + 8, 8)*7^n.
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 12082656/5 - 15676416*log(7/6).
Sum_{n>=0} (-1)^n/a(n) = 117440512*log(8/7) - 235229912/15. (End)

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A050982 5-idempotent numbers.

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A050988 6-idempotent numbers.

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A141054 8-idempotent numbers: a(n) = binomial(n+8,8)*8^n.

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A170932 a(n) = binomial(n + 8, 8)*7^n .

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