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.

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A108242 a(n) is the number of coverings of 1..n by cyclic words of length 3, such that each value from 1 to n appears precisely 3 times. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,1,2,2,2,...,n,n,n}. Repeats of words are allowed in a given covering.

Original entry on oeis.org

1, 1, 2, 16, 256, 7184, 311944, 19191448, 1584972224, 169021538944, 22595033625856, 3699135711988736, 727774085471066752, 169399730544125355136, 46039989792346454771456, 14447317177670702438831104, 5183889091511674280049885184, 2108937872584292649560886222848
Offset: 0

Views

Author

Marni Mishna, Jun 17 2005

Keywords

Comments

The asymptotic growth of the coefficients is a(n) ~ C (3/2)^n (n!)^2 /n with C approx 0.277.
In closed form, C = sqrt(3)/(2*Pi) = 0.27566444771089602475566324915648472... . - Vaclav Kotesovec, Feb 28 2016

Examples

			a(2)=2 because the two cyclic word coverings are {112, 221} and {111, 222}
a(3)=16: {111 222 333} {111 223 233} {112 122 333} {112 133 223} {113 122 233} {113 123 223} {113 132 223} {112 132 233} {113 133 222} {122 123 133} {122 132 133} {112 123 233} {123 123 123} {123 132 123} {123 132 132} {132 132 132}

Links

Crossrefs

Cf. A052502, A110105, A110106, A110104.

Programs

  • Mathematica
    RecurrenceTable[{-(-10 + n) (-9 + n) (-8 + n) (-7 + n) (-6 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) (25 - 243 n + 243 n^2) a[-11 + n] + 90 (-9 + n) (-8 + n) (-7 + n) (-6 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-10 + n] - 6 (-8 + n) (-7 + n) (-6 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) (52 - 270 n + 243 n^2) a[-9 + n] + 6 (-7 + n) (-6 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) (-40 + 1240 n - 1458 n^2 + 243 n^3) a[-8 + n] - (-6 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) (-917 - 3537 n + 3159 n^2) a[-7 + n] + 6 (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) (-711 + 4555 n - 4941 n^2 + 972 n^3) a[-6 + n] - 9 (-4 + n) (-3 + n) (-2 + n) (-1 + n) (-110 - 3557 n + 5128 n^2 - 1944 n^3 + 243 n^4) a[-5 + n] + 6 (-3 + n) (-2 + n) (-1 + n) (-508 + 4580 n - 5022 n^2 + 1215 n^3) a[-4 + n] - 6 (-2 + n) (-1 + n) (692 - 6471 n + 9309 n^2 - 4374 n^3 + 729 n^4) a[-3 + n] + 6 (-1 + n) (-92 + 2798 n - 3726 n^2 + 1215 n^3) a[-2 + n] - 3 (482 - 2451 n + 4206 n^2 - 2916 n^3 + 729 n^4) a[-1 + n] + 6 (511 - 729 n + 243 n^2) a[n] == 0, a[0] == 1, a[1] == 1, a[2] == 2, a[3] == 16, a[4] == 256, a[5] == 7184, a[6] == 311944, a[7] == 19191448, a[8] == 1584972224, a[9] == 169021538944, a[10] == 22595033625856}, a, {n, 0, 20}] (* Vaclav Kotesovec, Feb 28 2016 *)

Formula

Exponential generating function satisfies the linear differential equation: {(6 + 499*t^6 + 270*t^4 + 408*t^8 - 162*t^11 - 558*t^9 - 12*t - 96*t^3 + 66*t^2 - 654*t^7 + 60*t^12 + 154*t^10 - 342*t^5 + 9*t^14)*F(t) + (81*t^10 + 72*t^4 + 198*t^6 + 216*t^8 + 9*t^2)*(d^2/dt^2)F(t) + (-474*t^6 - 252*t^10 - 6 + 126*t^3 + 594*t^7 - 66*t^2 + 324*t^9 - 54*t^12 - 420*t^8 + 18*t - 264*t^4 + 378*t^5)*(d/dt)F(t), F(0) = 1}
The a(n) satisfy the recurrence: {a(0) = 1, a(1) = 1, ( - 20779902*n^7 - 134970693*n^6 - 1971620508*n^4 - 2248389*n^8 - 3*n^12 - 4459328640*n - 4242044664*n^3 - 5794678656*n^2 - 618210450*n^5 - 234*n^11 - 1437004800 - 8151*n^10 - 167310*n^9)*a(n) + ( - 7295434560*n - 4550515200 - 914850*n^7 - 5131406304*n^2 - 545289740*n^4 - 2088314700*n^3 - 11400627*n^6 - 95574465*n^5 - 1425*n^9 - 47310*n^8 - 19*n^10)*a(n + 2) + (711103032*n^4 + 8622028800 + 13032306*n^6 + 116250876*n^5 + 2944635984*n^3 + 12385923840*n + 7897844736*n^2 + 18*n^10 + 1404*n^9 + 48708*n^8 + 989496*n^7)*a(n + 3) + ( - 915980400*n - 898128000 - 3060*n^7 - 90090*n^6 - 1499400*n^5 - 15424605*n^4 - 100395540*n^3 - 403611660*n^2 - 45*n^8)*a(n + 4) + (2882376*n^5 + 890994600*n^2 + 2137510944*n + 30916662*n^4 + 210700728*n^3 + 166740*n^6 + 5472*n^7 + 78*n^8 + 2227357440)*a(n + 5) + ( - 1050477120 - 60979*n^6 - 1088733*n^5 - 12105088*n^4 - 27*n^8 - 85853091*n^3 - 379422466*n^2 - 955621272*n - 1944*n^7)*a(n + 6) + (57398400*n + 114*n^6 + 91238400 + 161430*n^4 + 2078100*n^3 + 14985456*n^2 + 6660*n^5)*a(n + 7) + ( - 1225827*n^3 - 58806000 - 63*n^6 - 9078336*n^2 - 92961*n^4 - 3753*n^5 - 35812260*n)*a(n + 8) + (571080*n + 1504800 + 5100*n^3 + 120*n^4 + 81060*n^2)*a(n + 9) + ( - 233178*n - 635976 - 32079*n^2 - 1962*n^3 - 45*n^4)*a(n + 10) + (1116*n + 48*n^2 + 6480)*a(n + 11) + ( - 225*n - 9*n^2 - 1410)*a(n + 12) + 6*a(n + 13) = 0,
with a(2) = 2, a(3) = 16, a(4) = 256, a(5) = 7184, a(6) = 311944, a(7) = 19191448, a(8) = 1584972224, a(9) = 169021538944, a(10) = 22595033625856, a(11) = 3699135711988736, a(12) = 727774085471066752}

Extensions

More terms from Vaclav Kotesovec, Feb 28 2016

A110104 a(n) is the number of coverings of 1..n by cyclic words of length 3n, such that each value from 1 to n appears precisely twice. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,2,2,...,n,n}. No repeats of words are allowed in a given covering.

Original entry on oeis.org

1, 4, 3760, 23504320, 567399078400, 37518268781593600, 5543744611870143078400, 1599334510537656091623424000, 818296434784062385011283591168000
Offset: 0

Views

Author

Marni Mishna, Jul 11 2005

Keywords

Comments

P-recursive.

Examples

			a(1)=4: {123, 132} {112, 233} {113, 322} {133, 122}

Crossrefs

Cf. A052502, A110105, A110106, A108242.

Programs

  • Mathematica
    RecurrenceTable[{(40320 + 328752*n + 1816668*n^3 + 1102248*n^5 + 398034*n^6 + 1818369*n^4 + 1063116*n^2 + 78732*n^7 + 6561*n^8) * a[n] + (508608*n + 161280 + 453600*n^3 + 34992*n^5 + 2916*n^6 + 173340*n^4 + 661104*n^2) * a[n + 1] + (12320 + 19980*n + 12096*n^2 + 3240*n^3 + 324*n^4) * a[n + 2] - 2*a[n + 3] == 0, a[0] == 1, a[1] == 4, a[2] == 3760}, a, {n, 0, 15}] (* Vaclav Kotesovec, Oct 24 2023 *)

Formula

Differential equation satisfied by egf: sum a(n)t^3n/(3n!) {F(0) = 1, (-2+4*t^6+16*t^3)*(d/dt)F(t) + 4*t^4*(d^2/dt^2)F(t) + t^2*(4+12*t^3+t^6)*F(t)}.
Recurrence: {a(0) = 1, (40320 + 328752*n + 1816668*n^3 + 1102248*n^5 + 398034*n^6 + 1818369*n^4 + 1063116*n^2 + 78732*n^7 + 6561*n^8)*a(n) +(508608*n + 161280 + 453600*n^3 + 34992*n^5 + 2916*n^6 + 173340*n^4 + 661104*n^2)*a(n+1) + (12320 + 19980*n + 12096*n^2 + 3240*n^3 + 324*n^4)*a(n+2) - 2*a(n+3), a(1) = 4, a(2) = 3760}.
a(n) ~ 2^n * 3^(4*n + 1/2) * n^(4*n) / exp(4*n). - Vaclav Kotesovec, Oct 24 2023

A319364 Expansion of e.g.f. exp(x^3/3)/(1 - x).

Original entry on oeis.org

1, 1, 2, 8, 32, 160, 1000, 7000, 56000, 506240, 5062400, 55686400, 668483200, 8690281600, 121663942400, 1825003980800, 29200063692800, 496401082777600, 8935231687782400, 169769402067865600, 3395388041357312000, 71303153503662080000, 1568669377080565760000, 36079395672853012480000
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2018

Keywords

Links

Crossrefs

Cf. A000090, A000522, A030977, A052502, A130905, A319365.

Programs

  • Maple
    seq(n!*coeff(series(exp(x^3/3)/(1 - x),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[x^3/3]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(x^3/3)/(1 - x))) \\ Michel Marcus, Dec 17 2020

Formula

a(n) ~ n!*exp(1/3).
D-finite with recurrence: n*a(n) - n^2*a(n-1) - n*(n-1)*(n-2)*a(n-3) + n*(n-1)*(n-2)*(n-3)*a(n-4) = 0. - Robert Israel, Dec 17 2020

A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 3, 0, 6, 1, 0, 0, 0, 24, 1, 15, 40, 0, 0, 120, 1, 0, 0, 0, 0, 0, 720, 1, 105, 0, 1260, 0, 0, 0, 5040, 1, 0, 2240, 0, 0, 0, 0, 0, 40320, 1, 945, 0, 0, 72576, 0, 0, 0, 0, 362880, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 1, 10395, 246400, 1247400, 0, 6652800, 0, 0, 0, 0, 0, 39916800
Offset: 1

Views

Author

Marko Riedel, Dec 17 2023

Keywords

Examples

			Row n=6 is 1, 15, 40, 120 because there is one permutation of [6] consisting of six fixed points, there are 15 permutations consisting of three transpositions, there are forty permutations consisting of two three-cycles and there are one hundred and twenty permutations consisting of just one six-cycle (6!/6).
Triangular array starts:
[ 1] 1;
[ 2] 1,   1;
[ 3] 1,   0,    2;
[ 4] 1,   3,    0,    6;
[ 5] 1,   0,    0,    0,    24;
[ 6] 1,  15,   40,    0,     0, 120;
[ 7] 1,   0,    0,    0,     0,   0, 720;
[ 8] 1, 105,    0, 1260,     0,   0,   0, 5040;
[ 9] 1,   0, 2240,    0,     0,   0,   0,    0, 40320;
[10] 1, 945,    0,    0, 72576,   0,   0,    0,     0, 362880;

References

  • P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, pages 120-122.

Links

Crossrefs

Cf. A005225 (row sums), A008290.
Cf. A123023 (column 2), A052502 (column 3), A060706 (column 4).

Programs

  • Maple
    T:= (n, m)-> `if`(irem(n,m)=0, n!/m^(n/m)/(n/m)!, 0):
seq(seq(T(n, m), m = 1..n), n=1..15);
  • Mathematica
    A368213[n_,k_]:=If[Divisible[n,k],n!/(k^(n/k)(n/k)!),0];
Table[A368213[n,k],{n,15},{k,n}] (* Paolo Xausa, Dec 18 2023 *)
  • SageMath
    def T(n, d): return factorial(n) // (d ** (n//d) * factorial(n//d))
for n in range(1, 19):
    print([T(n, d) if n % d == 0 else 0 for d in range(1, n+1)])
# Peter Luschny, Dec 17 2023

Formula

T(n, k) = n! / ( k^(n/k) * (n/k)! ) if k divides n otherwise 0.

A091753 Fourth column (m=5) of array A091752 ((-1,2)Stirling2) divided by -6.

Original entry on oeis.org

1, 60, 6720, 1232000, 336336000, 128076748800, 64892219392000, 42217023873024000, 34301331896832000000, 34042166278055936000000, 40523794737397786214400000, 56991191326140341157888000000, 93484550838645539612655616000000, 176901534663898482651640627200000000
Offset: 3

Views

Author

Wolfdieter Lang, Feb 27 2004

Keywords

Crossrefs

Cf. A052502, A091752.

Programs

  • Mathematica
    a[n_] := ((n-1)*(n-2)/4) * (3*(n-2))!/((3^(n-2))*(n-2)!); Array[a, 20, 3] (* Amiram Eldar, Sep 01 2025 *)

Formula

a(n) = -A091752(n, 5)/6, n>=3.
a(n) = ((n-1)*(n-2)/4)*(3*(n-2))!/((3^(n-2))*(n-2)!) = ((n-1)*(n-2)/4)*A052502(n-2), n>=3.
G.f.: (x^3)*hypergeom([4/3, 5/3, 3], [], 9*x).
a(n) ~ 3^(2*n-7/2) * n^(2*n-2) / (4 * exp(2*n-5/n)). - Amiram Eldar, Sep 01 2025

A091754 Fifth column (m=6) of array A091752 ((-1,2)Stirling2).

Original entry on oeis.org

1, 80, 9520, 1786400, 493292800, 189065676800, 96179539456000, 62739188255744000, 51070871935283200000, 50753775178192486400000, 60478693661116393062400000, 85121458839683971088384000000, 139713174879733993267265536000000, 264509913735543445488643604480000000
Offset: 3

Views

Author

Wolfdieter Lang, Feb 27 2004

Keywords

Crossrefs

Cf. A052502, A091752.

Programs

  • Mathematica
    a[n_] := ((3*(n-2))!/((3^(n-2))*(n-2)!))*(9*n^2-27*n+12)/4!; Array[a, 20, 3] (* Amiram Eldar, Sep 01 2025 *)

Formula

a(n) = A091752(n, 6), n>=3.
a(n) = ((3*(n-2))!/((3^(n-2))*(n-2)!))*(9*n^2-27*n+12)/4! = A052502(n-2)*(9*n^2-27*n+12)/4!, n>=3.
a(n) ~ (3/e)^(2*n-5/2) * n^(2*n-2) / 8. - Amiram Eldar, Sep 01 2025

A091755 Sixth column (m=7) of array A091752 ((-1,2)Stirling2) divided by -12.

Original entry on oeis.org

1, 140, 27720, 7847840, 3049446400, 1564366003200, 1026108219136000, 838477001922560000, 835580445006827520000, 997744946185930342400000, 1406513375677181496524800000, 2311431202054422682730496000000, 4380418953582248141850148864000000, 9483101601549384660012281888768000000
Offset: 4

Views

Author

Wolfdieter Lang, Feb 27 2004

Keywords

Crossrefs

Cf. A052502, A091752.

Programs

  • Mathematica
    a[n_] := (3/(4*5!))*n*(n-3)*(3*(n-2))!/((3^(n-2))*(n-2)!); Array[a, 15, 4] (* Amiram Eldar, Aug 30 2025 *)

Formula

a(n) = -A091752(n, 7)/12, n>=4.
a(n) = (3/(4*5!))*n*(n-3)*(3*(n-2))!/((3^(n-2))*(n-2)!) = 3*n*(n-3) * A052502(n-2)/(4*5!), n>=4.
G.f.: (x^4)*hypergeom([7/3, 8/3, 5, 2], [4], 9*x).
a(n) ~ 3^(2*n-7/2) * n^(2*n-2) /(160 * exp(2*n-5/n)). - Amiram Eldar, Aug 30 2025

A091756 Seventh column (m=8) of array A091752 ((-1,2)Stirling2).

Original entry on oeis.org

1, 220, 48720, 14463680, 5762556800, 3000655257600, 1987324218880000, 1634736979972096000, 1636859558116823040000, 1961447726093804748800000, 2772502956616965206835200000, 4565871212782705024303104000000, 8667353356325850744087642112000000, 18789301668434870837372923150336000000
Offset: 4

Views

Author

Wolfdieter Lang, Feb 27 2004

Keywords

Crossrefs

Cf. A052502, A091752.

Programs

  • Mathematica
    a[n_] := (9/6!)*(n-3)*(9*n^3 - 45*n^2 + 36*n + 40)*(3*(n-3))!/((3^(n-3))*(n-3)!); Array[a, 15, 4] (* Amiram Eldar, Aug 30 2025 *)

Formula

a(n) = A091752(n, 8), n>=4.
a(n) = (9/6!)*(n-3)*(9*n^3-45*n^2+36*n+40)*(3*(n-3))!/((3^(n-3))*(n-3)!) = (9/6!)*(n-3)*(9*n^3-45*n^2+36*n+40) * A052502(n-3), n>=4.
a(n) ~ 3^(2*n-7/2) * n^(2*n-2) /(80 * exp(2*n-12/n)). - Amiram Eldar, Aug 30 2025

A377597 Table read by antidiagonals: T(n,k) = (n*k)!/(n^k*k!), n >=1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 15, 40, 6, 1, 1, 105, 2240, 1260, 24, 1, 1, 945, 246400, 1247400, 72576, 120, 1, 1, 10395, 44844800, 3405402000, 1743565824, 6652800, 720, 1, 1, 135135, 12197785600, 19799007228000, 162193467211776, 4940103168000, 889574400, 5040, 1
Offset: 1

Views

Author

Peter Kagey, Nov 02 2024

Keywords

Comments

This is the number of permutations in S_{k*n} that consist of k disjoint n-cycles.

Examples

			The table begins:
n\k| 0  1     2          3               4                    5
---+-----------------------------------------------------------
 1 | 1  1     1          1               1                    1
 2 | 1  1     3         15             105                  945
 3 | 1  2    40       2240          246400             44844800
 4 | 1  6  1260    1247400      3405402000       19799007228000
 5 | 1 24 72576 1743565824 162193467211776 41363226782215962624
For example T(2,5) = (2*5)!/(2^5*5!) = 10!/(32*5!) = 945.

Crossrefs

Cf. A001147 (row 2), A052502 (row 3), A060706 (row 4), A052504 (row 5), A110468 (col 2).
Cf. A368213.
Previous Showing 11-19 of 19 results.

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