A050988 -id:A050988 - 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)

A140406 a(n) = binomial(n+6, 6)*8^n.

Original entry on oeis.org

1, 56, 1792, 43008, 860160, 15138816, 242221056, 3598712832, 50381979648, 671759728640, 8598524526592, 106309030510592, 1275708366127104, 14915974742409216, 170468282770391040, 1909244767028379648, 21001692437312176128, 227312435792084729856

Offset: 0

Zerinvary Lajos, Jun 16 2008

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (56,-1344,17920,-143360,688128,-1835008,2097152).

Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140405.

[8^n* Binomial(n+6, 6): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Maple
```
seq(binomial(n+6,6)*8^n,n=0..17);
```

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] (* Harvey P. Dale, Dec 15 2011 *)

a(n)=binomial(n+6,6)<<(3*n) \\ Charles R Greathouse IV, Dec 15 2011

G.f.: 1/(1-8*x)^7. - Zerinvary Lajos, Aug 06 2008
a(n) = 56*a(n-1) - 1344*a(n-2) + 17920*a(n-3) - 143360*a(n-4) + 688128*a(n-5) - 1835008*a(n-6) + 2097152*a(n-7). - Harvey P. Dale, Dec 15 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 538628/5 - 806736*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 2834352*log(9/8) - 1669188/5. (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)

A050989 7-idempotent numbers.

Original entry on oeis.org

1, 56, 1764, 41160, 792330, 13311144, 201885684, 2826399576, 37096494435, 461645264080, 5493578642552, 62926446269232, 697434779483988, 7510836086750640, 78863778910881720, 809668130151718992, 8147285559651672357, 80514351413028291528, 782778416515552834300

Offset: 7

Eric W. Weisstein

Vincenzo Librandi, Table of n, a(n) for n = 7..400
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (56,-1372,19208,-168070,941192,-3294172,6588344,-5764801).

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

[7^(n-7)* Binomial(n, 7): n in [7..30]]; // Vincenzo Librandi, Oct 16 2011

seq(binomial(n, 7)*7^(n-7), n=7..33); # Zerinvary Lajos, Aug 01 2008

LinearRecurrence[{56,-1372,19208,-168070,941192,-3294172,6588344,-5764801}, {1,56,1764,41160,792330,13311144,201885684,2826399576},20] (* Harvey P. Dale, May 31 2014 *)

a(n)=binomial(n, 7)*7^(n-7) \\ Charles R Greathouse IV, Sep 03 2011

a(n) = C(n, 7)*7^(n-7).
G.f.: x^7/(1-7*x)^8.
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=7} 1/a(n) = 2286144*log(7/6) - 10572289/30.
Sum_{n>=7} (-1)^(n+1)/a(n) = 12845056*log(8/7) - 51456517/30. (End)

A172501 a(n) = binomial(n+8,8)*6^n.

Original entry on oeis.org

1, 54, 1620, 35640, 641520, 10007712, 140107968, 1801388160, 21616657920, 244988789760, 2645878929408, 27420927086592, 274209270865920, 2657720625315840, 25058508752977920, 230538280527396864, 2074844524746571776, 18307451688940339200, 158664581304149606400

Offset: 0

Zerinvary Lajos, Feb 05 2010

With a different offset, number of n-permutations (n>=8) of 7 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..400
Index entries for linear recurrences with constant coefficients, signature (54,-1296,18144,-163296,979776,-3919104,10077696,-15116544,10077696).

Cf. A081136, A081144, A139626, A036084, A050988, A141407.

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

Table[Binomial[n + 8, 8]*6^n, {n, 0, 20}]

Vec(1 / (1 - 6*x)^9 + O(x^30)) \\ Colin Barker, Jul 24 2017

From Colin Barker, Jul 24 2017: (Start)
G.f.: 1 / (1 - 6*x)^9.
a(n) = (2^(-7 + n)*3^(-2 + n)*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n)) / 35.
(End)
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 4785948/7 - 3750000*log(6/5).
Sum_{n>=0} (-1)^n/a(n) = 39530064*log(7/6) - 213275484/35. (End)

A140405 a(n) = binomial(n+6, 6)*5^n.

Original entry on oeis.org

1, 35, 700, 10500, 131250, 1443750, 14437500, 134062500, 1173046875, 9775390625, 78203125000, 604296875000, 4532226562500, 33120117187500, 236572265625000, 1656005859375000, 11385040283203125, 77016448974609375, 513442993164062500, 3377914428710937500

Offset: 0

Zerinvary Lajos, Jun 16 2008

With a different offset, number of n-permutations (n>=6) of 6 objects: 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)=35 because we have
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.

Index entries for linear recurrences with constant coefficients, signature (35,-525,4375,-21875,65625,-109375,78125).

Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140406.

Maple
```
seq(binomial(n+6,6)*5^n,n=0..18);
```

Table[Binomial[n+6,6]5^n,{n,0,20}] (* Harvey P. Dale, Dec 03 2017 *)

G.f.: 1/(1-5*x)^7. - Zerinvary Lajos, Aug 06 2008
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 6856 - 30720*log(5/4).
Sum_{n>=0} (-1)^n/a(n) = 233280*log(6/5) - 42531. (End)

A173123 a(n) = binomial(n+9,9)*6^n.

Original entry on oeis.org

1, 60, 1980, 47520, 926640, 15567552, 233513280, 3202467840, 40831464960, 489977579520, 5585744406528, 60935393525760, 639821632020480, 6496650417438720, 64038411257610240, 614768748073058304, 5763457013184921600, 52888193768049868800, 475993743912448819200

Offset: 0

Zerinvary Lajos, Feb 10 2010

With a different offset, number of n-permutations (n>=9) of 7 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly 9 u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (60,-1620,25920,-272160,1959552,-9797760,33592320,-75582720,100776960,-60466176).

Cf. A081136, A081144, A139626, A036084, A050988, A141407, A172501.

[6^n* Binomial(n+9, 9): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 9, 9]*6^n, {n, 0, 20}]

a(n) = C(n + 9, 9)*6^n.
From Chai Wah Wu, Nov 12 2021: (Start)
a(n) = 60*a(n-1) - 1620*a(n-2) + 25920*a(n-3) - 272160*a(n-4) + 1959552*a(n-5) - 9797760*a(n-6) + 33592320*a(n-7) - 75582720*a(n-8) + 100776960*a(n-9) - 60466176*a(n-10) for n > 9.
G.f.: 1/(6*x - 1)^10. (End)
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 21093750*log(6/5) - 107683641/28.
Sum_{n>=0} (-1)^n/a(n) = 311299254*log(7/6) - 959739813/20. (End)

A173124 a(n) = binomial(n+10,10)*6^n.

Original entry on oeis.org

1, 66, 2376, 61776, 1297296, 23351328, 373621248, 5444195328, 73496636928, 930957401088, 11171488813056, 127964326404096, 1407607590445056, 14942295960109056, 153692187018264576, 1536921870182645760, 14984988234280796160, 142798123173734645760, 1332782482954856693760

Offset: 0

Zerinvary Lajos, Feb 10 2010

With a different offset, number of n-permutations (n>=10) of 7 objects: r, s, t, u, v, z, x, with repetition allowed, containing exactly 10 u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (66,-1980,35640,-427680,3592512,-21555072,92378880,-277136640,554273280,-665127936,362797056).

Cf. A081136, A081144, A139626, A036084, A050988, A141407, A172501, A173123.

[6^n* Binomial(n+10, 10): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 10, 10]*6^n, {n, 0, 20}]

From Chai Wah Wu, Nov 12 2021: (Start)
a(n) = 66*a(n-1) - 1980*a(n-2) + 35640*a(n-3) - 427680*a(n-4) + 3592512*a(n-5) - 21555072*a(n-6) + 92378880*a(n-7) - 277136640*a(n-8) + 554273280*a(n-9) - 665127936*a(n-10) + 362797056*a(n-11) for n > 10.
G.f.: -1/(6*x - 1)^11. (End)
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=0} 1/a(n) = 897363955/42 - 117187500*log(6/5).
Sum_{n>=0} (-1)^n/a(n) = 2421216420*log(7/6) - 2239392937/6. (End)

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A050982 5-idempotent numbers.

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A140406 a(n) = binomial(n+6, 6)*8^n.

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

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A050989 7-idempotent numbers.

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A172501 a(n) = binomial(n+8,8)*6^n.

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A140405 a(n) = binomial(n+6, 6)*5^n.

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A173123 a(n) = binomial(n+9,9)*6^n.

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