A185105 -id:A185105 - cp's OEIS frontend

A285233 Number of entries in the fifth cycles of all permutations of [n].

1, 17, 221, 2724, 34009, 441383, 6020276, 86673088, 1318681308, 21194234508, 359421505224, 6421154849208, 120637782989568, 2379195625677696, 49167226489281408, 1062833010282628992, 23992442301958329600, 564697104190192569600, 13836823816466433139200

Offset: 5

Alois P. Heinz, Apr 15 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Alois P. Heinz, Table of n, a(n) for n = 5..449
Wikipedia, Permutation

Column k=5 of A185105.

a:= proc(n) option remember; `if`(n<6, [0$5, 1][n+1],
      ((4*(n^3-9*n^2+24*n-19))*a(n-1)-(6*n^4-72*n^3+
       307*n^2-547*n+334)*a(n-2)+(4*n^5-64*n^4+398*n^3
      -1191*n^2+1683*n-862)*a(n-3)-(n-4)^5*(n-1)*a(n-4))
      /((n-2)*(n-5)))
    end:
seq(a(n), n=5..25);

a[3] = a[4] = 0; a[5] = 1; a[6] = 17; a[n_] := a[n] = ((4(n^3 - 9n^2 + 24n - 19)) a[n-1] - (6n^4 - 72n^3 + 307n^2 - 547n + 334) a[n-2] + (4n^5 - 64n^4 + 398n^3 - 1191n^2 + 1683n - 862) a[n-3] - (n-4)^5 (n-1) a[n-4]) / ((n - 2)(n - 5));
Table[a[n], {n, 5, 25}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) = A185105(n,5).

a(n) ~ n!*n/32. - Vaclav Kotesovec, Apr 25 2017

A285234 Number of entries in the sixth cycles of all permutations of [n].

Original entry on oeis.org

1, 23, 382, 5780, 86029, 1301673, 20338679, 330737236, 5618265376, 99849949772, 1857170751804, 36135886878072, 734947859916792, 15608257104179952, 345724111468700496, 7977315239656638912, 191516062334747746752, 4778050475554642998144, 123731984754223222096512

Offset: 6

Alois P. Heinz, Apr 15 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Alois P. Heinz, Table of n, a(n) for n = 6..449
Wikipedia, Permutation

Column k=6 of A185105.

a:= proc(n) option remember; `if`(n<7, [0$6, 1][n+1],
      ((5*n^3-58*n^2+207*n-230)*a(n-1)-(10*n^4-152*n^3
       +835*n^2-1973*n+1690)*a(n-2)+(n-4)*(10*n^4
       -158*n^3+909*n^2-2251*n+2000)*a(n-3)-(5*n^6
       -127*n^5+1330*n^4-7335*n^3+22396*n^2-35717*n
       +23058)*a(n-4)+(n-5)^6*(n-2)*a(n-5))/((n-3)*(n-6)))
    end:
seq(a(n), n=6..25);

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[Function[p, p + Coefficient[ p, x, 0]*j*x^i][b[n-j, i+1]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]]];
a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]][[6]];
Table[a[n], {n, 6, 25}] (* Jean-François Alcover, Jun 01 2018, after Alois P. Heinz *)

a(n) = A185105(n,6).

a(n) ~ n!*n/64. - Vaclav Kotesovec, Apr 25 2017

A285235 Number of entries in the seventh cycles of all permutations of [n].

Original entry on oeis.org

1, 30, 622, 11378, 199809, 3499572, 62333543, 1141073295, 21593291506, 423749322362, 8637159909596, 182967605341204, 4028364756058464, 92147187469290768, 2188667860854515856, 53939340317601471888, 1378181549321980128288, 36476226109960185948768

Offset: 7

Alois P. Heinz, Apr 15 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Alois P. Heinz, Table of n, a(n) for n = 7..450
Wikipedia, Permutation

Column k=7 of A185105.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
      b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
      (j-1)!, j=1..n)))
    end:
a:= n-> coeff(b(n, 7), x, 1):
seq(a(n), n=7..30);

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[Function[p, p + If[i == 1, Coefficient[p, x, 0]*j*x, 0]][b[n - j, Max[0, i - 1]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]]];
a[n_] := Coefficient[b[n, 7], x, 1];
Table[a[n], {n, 7, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) = A185105(n,7).
Recurrence: (n-7)*(n-4)*a(n) = (n-3)*(6*n^2 - 67*n + 176)*a(n-1) - 5*(n-4)*(3*n^3 - 43*n^2 + 195*n - 283)*a(n-2) + 10*(2*n^5 - 47*n^4 + 436*n^3 - 1999*n^2 + 4532*n - 4062)*a(n-3) - (15*n^6 - 445*n^5 + 5465*n^4 - 35555*n^3 + 129161*n^2 - 248111*n + 196528)*a(n-4) + (6*n^7 - 221*n^6 + 3473*n^5 - 30165*n^4 + 156251*n^3 - 482105*n^2 + 819087*n - 589808)*a(n-5) - (n-6)^7*(n-3)*a(n-6), for n>7. - Vaclav Kotesovec, Apr 25 2017
a(n) ~ n!*n/128. - Vaclav Kotesovec, Apr 25 2017

A285236 Number of entries in the eighth cycles of all permutations of [n].

Original entry on oeis.org

1, 38, 964, 21018, 431007, 8671656, 175065071, 3591984289, 75473055841, 1631318215818, 36369569578502, 837619857754240, 19943142053389024, 491010028537071248, 12499878460133012064, 328936666440527737296, 8943724877454118086096, 251125623168859020015072

Offset: 8

Alois P. Heinz, Apr 15 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Alois P. Heinz, Table of n, a(n) for n = 8..450
Wikipedia, Permutation

Column k=8 of A185105.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
      b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
      (j-1)!, j=1..n)))
    end:
a:= n-> coeff(b(n, 8), x, 1):
seq(a(n), n=8..30);

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[Function[p, p + If[i == 1, Coefficient[p, x, 0]*j*x, 0]][b[n - j, Max[0, i - 1]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]]];
a[n_] := Coefficient[b[n, 8], x, 1];
Table[a[n], {n, 8, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) = A185105(n,8).
Recurrence: (n-8)*(n-5)*a(n) = (7*n^3 - 117*n^2 + 618*n - 1036)*a(n-1) - (21*n^4 - 450*n^3 + 3521*n^2 - 11996*n + 15092)*a(n-2) + 5*(7*n^5 - 190*n^4 + 2039*n^3 - 10842*n^2 + 28614*n - 30016)*a(n-3) - (35*n^6 - 1185*n^5 + 16635*n^4 - 124015*n^3 + 518011*n^2 - 1149493*n + 1058400)*a(n-4) + (n-6)*(21*n^6 - 747*n^5 + 11033*n^4 - 86597*n^3 + 380805*n^2 - 888917*n + 859586)*a(n-5) - (7*n^8 - 352*n^7 + 7728*n^6 - 96726*n^5 + 754656*n^4 - 3756732*n^3 + 11646888*n^2 - 20547489*n + 15780868)*a(n-6) + (n-7)^8*(n-4)*a(n-7), for n>8. - Vaclav Kotesovec, Apr 25 2017
a(n) ~ n!*n/256. - Vaclav Kotesovec, Apr 25 2017

A285237 Number of entries in the ninth cycles of all permutations of [n].

Original entry on oeis.org

1, 47, 1434, 36792, 872511, 20014299, 455265257, 10420963144, 242208466145, 5748862140283, 139849088103596, 3494752531722564, 89838192687840304, 2377612074981717632, 64807344109730799968, 1819505580964336136560, 52611858820598185363536

Offset: 9

Alois P. Heinz, Apr 15 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Alois P. Heinz, Table of n, a(n) for n = 9..450
Wikipedia, Permutation

Column k=9 of A185105.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
      b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
      (j-1)!, j=1..n)))
    end:
a:= n-> coeff(b(n, 9), x, 1):
seq(a(n), n=9..30);

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[Function[p, p + If[i == 1, Coefficient[p, x, 0]*j*x, 0]][b[n - j, Max[0, i - 1]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]]];
a[n_] := Coefficient[b[n, 9], x, 1];
Table[a[n], {n, 9, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) = A185105(n,9).
Recurrence: (n-9)*(n-6)*a(n) = 2*(4*n^3 - 77*n^2 + 471*n - 916)*a(n-1) - 14*(2*n^4 - 49*n^3 + 439*n^2 - 1714*n + 2474)*a(n-2) + 14*(n-5)*(4*n^4 - 103*n^3 + 981*n^2 - 4117*n + 6454)*a(n-3) - 7*(n-6)*(10*n^5 - 320*n^4 + 4070*n^3 - 25770*n^2 + 81333*n - 102427)*a(n-4) + 14*(4*n^7 - 185*n^6 + 3661*n^5 - 40195*n^4 + 264477*n^3 - 1042986*n^2 + 2282488*n - 2138058)*a(n-5) - (28*n^8 - 1554*n^7 + 37702*n^6 - 522242*n^5 + 4517128*n^4 - 24979724*n^3 + 86233855*n^2 - 169871843*n + 146155098)*a(n-6) + (8*n^9 - 526*n^8 + 15356*n^7 - 261226*n^6 + 2853242*n^5 - 20747608*n^4 + 100420076*n^3 - 311890495*n^2 + 563892963*n - 452026202)*a(n-7) - (n-8)^9*(n-5)*a(n-8), for n>9. - Vaclav Kotesovec, Apr 25 2017
a(n) ~ n!*n/512. - Vaclav Kotesovec, Apr 25 2017

A285238 Number of entries in the tenth cycles of all permutations of [n].

Original entry on oeis.org

1, 57, 2061, 61524, 1672323, 43426821, 1106667572, 28127644296, 720378419177, 18715673685469, 495446888127507, 13403690294272704, 371315688867567088, 10546557230068193568, 307378160401299252032, 9196581430595518185328, 282526394585486139996736

Offset: 10

Alois P. Heinz, Apr 15 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Alois P. Heinz, Table of n, a(n) for n = 10..450
Wikipedia, Permutation

Column k=10 of A185105.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
      b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
      (j-1)!, j=1..n)))
    end:
a:= n-> coeff(b(n, 10), x, 1):
seq(a(n), n=10..30);

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[Function[p, p + If[i == 1, Coefficient[p, x, 0]*j*x, 0]][b[n - j, Max[0, i - 1]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]]];
a[n_] := Coefficient[b[n, 10], x, 1];
Table[a[n], {n, 10, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) = A185105(n,10).
Recurrence: (n-10)*(n-7)*a(n) = (9*n^3 - 196*n^2 + 1361*n - 3006)*a(n-1) - 2*(18*n^4 - 496*n^3 + 5001*n^2 - 21971*n + 35682)*a(n-2) + 14*(6*n^5 - 206*n^4 + 2797*n^3 - 18829*n^2 + 63002*n - 83988)*a(n-3) - 7*(18*n^6 - 758*n^5 + 13240*n^4 - 122950*n^3 + 640883*n^2 - 1779393*n + 2057142)*a(n-4) + 7*(18*n^7 - 916*n^6 + 19950*n^5 - 241180*n^4 + 1748577*n^3 - 7604998*n^2 + 18375843*n - 19031658)*a(n-5) - (84*n^8 - 5096*n^7 + 135226*n^6 - 2050286*n^5 + 19428976*n^4 - 117838826*n^3 + 446719463*n^2 - 967742093*n + 917171710)*a(n-6) + (n-8)*(36*n^8 - 2284*n^7 + 63390*n^6 - 1005202*n^5 + 9960732*n^4 - 63153820*n^3 + 250166217*n^2 - 565970839*n + 559792740)*a(n-7) - (9*n^10 - 749*n^9 + 28038*n^8 - 621666*n^7 + 9040584*n^6 - 90096090*n^5 + 623077092*n^4 - 2952338109*n^3 + 9171809128*n^2 - 16867010733*n + 13941384550)*a(n-8) + (n-9)^10*(n-6)*a(n-9), for n>10. - Vaclav Kotesovec, Apr 25 2017
a(n) ~ n!*n/1024. - Vaclav Kotesovec, Apr 25 2017

cp's OEIS Frontend

A285233 Number of entries in the fifth cycles of all permutations of [n].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A285234 Number of entries in the sixth cycles of all permutations of [n].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A285235 Number of entries in the seventh cycles of all permutations of [n].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A285236 Number of entries in the eighth cycles of all permutations of [n].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A285237 Number of entries in the ninth cycles of all permutations of [n].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A285238 Number of entries in the tenth cycles of all permutations of [n].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula