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 1-6 of 6 results.

A159716 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

2, 66, 1168, 16220, 202416, 2395540, 27517568, 310123764, 3447919120, 37934904788, 413863668480, 4483624403284, 48285543009872, 517346347249140, 5518365322864384, 58632646191319220, 620816303380261392, 6553061146974071956, 68979591578665208960, 724285713430953995412
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column k=2 of A334772.

Programs

  • PARI
    a(n) = {n*(169*10^(n-2) + 48*3^(n-2) - 84*n*3^(n-2))/49} \\ Andrew Howroyd, May 10 2020
  • PARI
    Vec(2*x*(1 + 3*x)*(1 + x - 69*x^2 + 45*x^3) / ((1 - 3*x)^3*(1 - 10*x)^2) + O(x^20)) \\ Colin Barker, May 19 2020

Formula

a(n) = n*(169*10^(n-2) + 48*3^(n-2) - 84*n*3^(n-2))/49. - Andrew Howroyd, May 10 2020
From Colin Barker, May 19 2020: (Start)
G.f.: 2*x*(1 + 3*x)*(1 + x - 69*x^2 + 45*x^3) / ((1 - 3*x)^3*(1 - 10*x)^2).
a(n) = 29*a(n-1) - 307*a(n-2) + 1467*a(n-3) - 3240*a(n-4) + 2700*a(n-5) for n>6.
(End)

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 09 2020

A334773 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

3, 12, 57, 30, 360, 705, 60, 1400, 7968, 7617, 105, 4170, 51750, 163584, 78357, 168, 10437, 241080, 1830000, 3293184, 791589, 252, 23072, 894201, 13562040, 64168750, 65968128, 7944321, 360, 46440, 2804480, 75278553, 759940800, 2246625000, 1319854080, 79541625
Offset: 2

Views

Author

Andrew Howroyd, May 10 2020

Keywords

Examples

			Array begins:
======================================================
n\k |       2          3           4              5
----|-------------------------------------------------
  2 |       3         12          30            60 ...
  3 |      57        360        1400          4170 ...
  4 |     705       7968       51750        241080 ...
  5 |    7617     163584     1830000      13562040 ...
  6 |   78357    3293184    64168750     759940800 ...
  7 |  791589   65968128  2246625000   42560067360 ...
  8 | 7944321 1319854080 78636093750 2383387566720 ...
  ...
The T(2,2) = 3 permutations of 1122 with 2 local maxima are 1212, 2112, 2121.

Links

Crossrefs

Columns k=2..8 are 3*A152494, 12*A152499, 10*A152504, 30*A152509, 21*A152513, 56*A152517, 36*A152518.
Cf. A334772, A334774, A334778.

Programs

  • PARI
    T(n,k) = {3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2}

Formula

T(n,k) = Sum_{j=0..n-2} P(k-1,3) * P(k-2,2) * P(k,2)^(n-2-j) * P(k,4)^j + 2 * (n-j-2) * P(k-1,3)^2 * P(k,2)^(n-3-j) * P(k,4)^j where P(n,k) = binomial(n+k-1,k-1).
T(n,k) = 3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2.

A159722 Number of permutations of 3 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

12, 576, 17376, 448800, 10861056, 253940736, 5807161344, 130675728384, 2903978803200, 63887897001984, 1393919508086784, 30201597684350976, 650495989232173056, 13939199950454784000, 297369599774111563776, 6319103998978368208896, 133816319995412169621504
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column k=3 of A334772.
Cf. A159716.

Programs

  • PARI
    a(n) = {3*n*(121*20^(n-2) + 15*4^(n-2) - 36*n*4^(n-2))/32} \\ Andrew Howroyd, May 10 2020
  • PARI
    Vec(12*x*(1 + 2*x)*(1 - 6*x - 108*x^2 + 80*x^3) / ((1 - 4*x)^3*(1 - 20*x)^2) + O(x^20)) \\ Colin Barker, Jul 16 2020

Formula

a(n) = 3*n*(121*20^(n-2) + 15*4^(n-2) - 36*n*4^(n-2))/32. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 16 2020: (Start)
G.f.: 12*x*(1 + 2*x)*(1 - 6*x - 108*x^2 + 80*x^3) / ((1 - 4*x)^3*(1 - 20*x)^2).
a(n) = 52*a(n-1) - 928*a(n-2) + 6784*a(n-3) - 21760*a(n-4) + 25600*a(n-5) for n>6.
(End)

Extensions

Terms a(9) and beyond from Andrew Howroyd, May 09 2020

A159728 Number of permutations of 4 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

36, 2610, 129800, 5748750, 241987500, 9885006250, 395426250000, 15570077343750, 605504070312500, 23311913238281250, 890091272109375000, 33749294301074218750, 1272088786561523437500, 47703329503967285156250, 1780924301526757812500000, 66228122463283630371093750
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column k=4 of A334772.
Cf. A159716.

Programs

  • PARI
    a(n) = {2*n*(121*35^(n-2) + 8*5^(n-2) - 24*n*5^(n-2))/9} \\ Andrew Howroyd, May 10 2020
  • PARI
    Vec(2*x^2*(3 + 5*x)*(6 - 85*x - 1100*x^2 + 875*x^3) / ((1 - 5*x)^3*(1 - 35*x)^2) + O(x^40)) \\ Colin Barker, Jul 16 2020

Formula

a(n) = 2*n*(121*35^(n-2) + 8*5^(n-2) - 24*n*5^(n-2))/9. - Andrew Howroyd, May 10 2020

Extensions

Terms a(8) and beyond from Andrew Howroyd, May 09 2020

A159734 Number of permutations of 5 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

80, 8520, 659560, 46412200, 3121135440, 203933233280, 13051880894720, 822269693093760, 51163456598214400, 3151668992962800640, 192538324414433556480, 11680658351228331345920, 704433549821153777192960, 42266012989435750480281600, 2524689842570106278817955840
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column k=5 of A334772.
Cf. A159716.

Programs

  • PARI
    a(n) = {n*(23^2*56^(n-2) + 21*6^(n-2) - 75*n*6^(n-2))/10} \\ Andrew Howroyd, May 10 2020
  • PARI
    Vec(40*x^2*(2 + 3*x)*(1 - 25*x - 303*x^2 + 252*x^3) / ((1 - 6*x)^3*(1 - 56*x)^2) + O(x^18)) \\ Colin Barker, Jul 16 2020

Formula

a(n) = n*(23^2*56^(n-2) + 21*6^(n-2) - 75*n*6^(n-2))/10. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 16 2020: (Start)
G.f.: 40*x^2*(2 + 3*x)*(1 - 25*x - 303*x^2 + 252*x^3) / ((1 - 6*x)^3*(1 - 56*x)^2).
a(n) = 130*a(n-1) - 5260*a(n-2) + 68760*a(n-3) - 362880*a(n-4) + 677376*a(n-5) for n>6.
(End)

Extensions

Terms a(7) and beyond from Andrew Howroyd, May 09 2020

A159737 Number of permutations of 6 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

150, 22680, 2596608, 273322980, 27558217008, 2700777267972, 259275295383552, 24501521550788100, 2286808732032093360, 211301127303186249252, 19362866942233277773632, 1762020891775616889450852, 159395120671659354639719856, 14345560860451487040265198020
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column k=6 of A334772.
Cf. A159716.

Programs

  • PARI
    a(n) = {3*n*(61^2*84^(n-2) + 96*7^(n-2) - 396*n*7^(n-2))/121} \\ Andrew Howroyd, May 10 2020
  • PARI
    Vec(6*x^2*(5 + 7*x)*(5 - 196*x - 2401*x^2 + 2058*x^3) / ((1 - 7*x)^3*(1 - 84*x)^2) + O(x^40)) \\ Colin Barker, Jul 18 2020

Formula

a(n) = 3*n*(61^2*84^(n-2) + 96*7^(n-2) - 396*n*7^(n-2))/121. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 18 2020: (Start)
G.f.: 6*x^2*(5 + 7*x)*(5 - 196*x - 2401*x^2 + 2058*x^3) / ((1 - 7*x)^3*(1 - 84*x)^2).
a(n) = 189*a(n-1) - 10731*a(n-2) + 173215*a(n-3) - 1094856*a(n-4) + 2420208*a(n-5) for n>6.
(End)

Extensions

Terms a(7) and beyond from Andrew Howroyd, May 09 2020
Showing 1-6 of 6 results.

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