A140406 -id:A140406 - cp's OEIS frontend

A053107 Expansion of 1/(1-8*x)^8.

1, 64, 2304, 61440, 1351680, 25952256, 449839104, 7197425664, 107961384960, 1535450808320, 20882130993152, 273366078455808, 3462636993773568, 42617070692597760, 511404848311173120, 6000483553517764608, 69005560865454292992, 779356922715719073792

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n>=7) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly 7 u's. - Zerinvary Lajos, Feb 11 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (64, -1792, 28672, -286720, 1835008, -7340032, 16777216, -16777216).

Cf. A036226.

Cf. A081138, A140802, A175210, A140406, A053107, A141054, A173155. - Zerinvary Lajos, Feb 11 2010

[8^n* Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Table[Binomial[n + 7, 7]*8^n, {n, 0, 20}] (* Zerinvary Lajos, Feb 11 2010 *)
CoefficientList[Series[1/(1-8x)^8,{x,0,20}],x] (* or *) LinearRecurrence[ {64,-1792,28672,-286720,1835008,-7340032,16777216,-16777216},{1,64,2304,61440,1351680,25952256,449839104,7197425664},20] (* Harvey P. Dale, Jul 19 2018 *)

vector(30, n, n--; 8^n*binomial(n+7,7)) \\ G. C. Greubel, Aug 16 2018

[lucas_number2(n, 8, 0)*binomial(n,7)/8^7 for n in range(7, 22)] # Zerinvary Lajos, Mar 13 2009

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

G.f.: 1/(1-8*x)^8.

More terms from Harvey P. Dale, Jul 19 2018

A173155 a(n) = binomial(n + 5, 5) * 8^n.

Original entry on oeis.org

1, 48, 1344, 28672, 516096, 8257536, 121110528, 1660944384, 21592276992, 268703891456, 3224446697472, 37520834297856, 425236122042368, 4710307813392384, 51140484831117312, 545498504865251328, 5727734301085138944, 59298896293587320832, 606166495445559279616

Offset: 0

Zerinvary Lajos, Feb 11 2010

With a different offset, number of n-permutations (n>=5) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly five (5) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (48,-960,10240,-61440,196608,-262144).

Cf. A081138, A140802, A175210, A140406, A053107, A141054.

[8^n* Binomial(n+5, 5): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Table[Binomial[n + 5, 5]*8^n, {n, 0, 20}]

a(n) = C(n + 5, 5)*8^n, n>=0.
G.f.: 1/(1-8*x)^6. - Vincenzo Librandi, Oct 16 2011
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 96040*log(8/7) - 38470/3.
Sum_{n>=0} (-1)^n/a(n) = 262440*log(9/8) - 30910. (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)

A172510 a(n) = binomial(n + 4, 4) * 8^n.

Original entry on oeis.org

1, 40, 960, 17920, 286720, 4128768, 55050240, 692060160, 8304721920, 95965675520, 1074815565824, 11725260718080, 125069447659520, 1308418837053440, 13458022323978240, 136374626216312832, 1363746262163128320, 13477021884906209280, 131775325096860712960

Offset: 0

Zerinvary Lajos, Feb 05 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..138
Index entries for linear recurrences with constant coefficients, signature (40,-640,5120,-20480,32768).

Cf. A081138, A140802, A140406, A053107, A141054.

[Binomial(n + 4, 4)*8^n: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

Table[Binomial[n + 4, 4]*8^n, {n, 0, 25}]

Vec(1 / (1-8*x)^5 + O(x^30)) \\ Colin Barker, Jul 24 2017

G.f.: 1 / (1-8*x)^5. - R. J. Mathar, Feb 11 2010
a(n) = (8^(-1 + n)*(1 + n)*(2 + n)*(3 + n)*(4 + n)) / 3. - Colin Barker, Jul 24 2017
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 4400/3 - 10976*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 23328*log(9/8) - 8240/3. (End)

A196280 a(n) = binomial(n+9, 9)*8^n.

Original entry on oeis.org

1, 80, 3520, 112640, 2928640, 65601536, 1312030720, 23991418880, 407854120960, 6525665935360, 99190122217472, 1442765414072320, 20198715797012480, 273459536944168960, 3594039628409077760, 46003707243636195328, 575046340545452441600, 7035861107850241638400

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (80,-2880,61440,-860160,8257536,-55050240,251658240,-754974720,1342177280,-1073741824).

Cf. A140406, A140802, A141054, A173155

Magma
```
[Binomial(n+9, 9)*8^n: n in [0..20]];
```

Table[Binomial[n+9,9]8^n,{n,0,20}] (* or *) LinearRecurrence[{80,-2880,61440,-860160,8257536,-55050240,251658240,-754974720,1342177280,-1073741824},{1,80,3520,112640,2928640,65601536,1312030720,23991418880,407854120960,6525665935360},20] (* Harvey P. Dale, May 13 2017 *)

a(n) = C(n+9,9)*8^n.
G.f.: 1 / (8*x-1)^10 . - R. J. Mathar, Oct 13 2011
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=0} 1/a(n) = 415065672*log(8/7) - 277121481/5.
Sum_{n>=0} (-1)^n/a(n) = 3099363912*log(9/8) - 12776837121/35. (End)

A197321 a(n) = binomial(n+10, 10)*8^n.

Original entry on oeis.org

1, 88, 4224, 146432, 4100096, 98402304, 2099249152, 40785412096, 734137417728, 12398765277184, 198380244434944, 3029807369551872, 44437174753427456, 628956934971588608, 8625695108181786624, 115009268109090488320, 1495120485418176348160, 18996824991195652423680

Offset: 0

Vincenzo Librandi, Oct 15 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (88,-3520,84480,-1351680,15138816,-121110528,692060160,-2768240640,7381975040,-11811160064,8589934592).

Cf. A140406, A140802, A141054, A173155, A196280.

Magma
```
[8^n*Binomial(n+10, 10): n in [0..20]]
```

Table[Binomial[n+10,10]8^n,{n,0,20}] (* Harvey P. Dale, Mar 05 2012 *)

a(n) = 8^n*C(n+10, 10).
G.f.: 1/(1-8*x)^11.
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=0} 1/a(n) = 3879700814/9 - 3228288560*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 30993639120*log(9/8) - 229983068738/63. (End)

cp's OEIS Frontend

A053107 Expansion of 1/(1-8*x)^8.

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A173155 a(n) = binomial(n + 5, 5) * 8^n.

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

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A172510 a(n) = binomial(n + 4, 4) * 8^n.

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A196280 a(n) = binomial(n+9, 9)*8^n.

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A197321 a(n) = binomial(n+10, 10)*8^n.

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