cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Zerinvary Lajos, Feb 05 2010

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Magma
    [6^n* Binomial(n+8, 8): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011
  • Mathematica
    Table[Binomial[n + 8, 8]*6^n, {n, 0, 20}]
  • PARI
    Vec(1 / (1 - 6*x)^9 + O(x^30)) \\ Colin Barker, Jul 24 2017

Formula

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)

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

Views

Author

Zerinvary Lajos, Feb 10 2010

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Zerinvary Lajos, Feb 10 2010

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Formula

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)
Showing 1-3 of 3 results.

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