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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A239839 Number of ordered pairs of permutation functions on n elements satisfying f(f(f(x))) = g(f(g(x))).

Original entry on oeis.org

1, 1, 4, 18, 168, 1560, 20880, 267120, 5080320, 93623040, 2184537600, 49896000000, 1451853849600, 41739720422400, 1426847092070400, 47989033956864000, 1919268439216128000, 76229151152394240000, 3471527082588364800000, 156226856133456396288000
Offset: 0

Views

Author

Chad Brewbaker, Mar 27 2014

Keywords

Links

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

Extensions

a(8)-a(9) from Giovanni Resta, Mar 27 2014
a(10)-a(13) from Paul Boddington, Feb 23 2015
a(14)-a(19) from Hiroaki Yamanouchi, Mar 12 2015

A239840 Number of ordered pairs of permutation functions (f,g) on n elements satisfying f(x) = f(g(g(x))).

Original entry on oeis.org

1, 1, 4, 24, 240, 3120, 54720, 1169280, 30804480, 950745600, 34459084800, 1424870092800, 67133032243200, 3540086232883200, 208397961547776000, 13533822947893248000, 966773828738285568000, 75334352557782269952000, 6385175803136642383872000
Offset: 0

Views

Author

Chad Brewbaker, Mar 27 2014

Keywords

Links

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

Programs

  • Maple
    a:= proc(n) a(n):= `if`(n<2, 1, n*a(n-1) +n*(n-1)^2*a(n-2)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2014
  • Mathematica
    a[n_] := a[n] = n a[n-1] + n(n-1)^2 a[n-2]; a[0] = a[1] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Oct 04 2019 *)

Formula

From Alois P. Heinz, Jul 23 2014: (Start)
a(n) = n! * A000085(n) = A000142(n) * A000085(n).
a(n) = n*a(n-1) + n*(n-1)^2*a(n-2) for n>=2, a(0) = a(1) = 1. (End)
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x + x^2 / 2). - Ilya Gutkovskiy, Jul 15 2021

Extensions

a(8)-a(9) from Giovanni Resta, Mar 27 2014
a(10)-a(18) from Alois P. Heinz, Jul 23 2014

A254529 a(n) = n! * (number of mapping patterns on n).

Original entry on oeis.org

1, 1, 6, 42, 456, 5640, 93600, 1728720, 38344320, 948931200, 26555558400, 817935148800, 27735629644800, 1020596255078400, 40642432179148800, 1737890081351424000, 79498734605402112000, 3871319396080840704000, 200017645344178421760000, 10925549584125028909056000
Offset: 0

Views

Author

Martin Fuller, Feb 01 2015

Keywords

Comments

a(n) is the number of ordered pairs (p, f) such that p f = f p, where p is a permutation and f is an endofunction.

Links

Crossrefs

Cf. A001372, A053529, A181162.

Formula

a(n) = n! * A001372(n). - Joerg Arndt, Feb 01 2015

A330200 Expansion of e.g.f. Product_{k>=1} exp(x^k) / (1 - x^k).

Original entry on oeis.org

1, 2, 9, 52, 389, 3366, 34477, 392624, 5035977, 70674634, 1085687921, 17982460332, 321298513549, 6121639481582, 124336400707989, 2674237637496616, 60799325536137617, 1454405117742700434, 36556297436871331417, 961899014831786663204
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2019

Keywords

Crossrefs

Cf. A000010, A000203, A000262, A000522, A053529, A330201, A346964.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Product[Exp[x^k]/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (n - 1)! Sum[(DivisorSigma[1, k] + k) a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[n!*Sum[LaguerreL[k, -1, -1]*PartitionsP[n-k],{k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 09 2021 *)

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of A000522.
E.g.f.: exp(Sum_{k>=1} (sigma(k) / k + 1) * x^k), where sigma = A000203.
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(phi(k)/k + 1), where phi = A000010.
a(0) = 1; a(n) = (n - 1)! * Sum_{k=1..n} (sigma(k) + k) * a(n-k) / (n - k)!.
a(n) = Sum_{k=0..n} binomial(n,k) * A000262(k) * A053529(n-k).
a(n) ~ sqrt(1/Pi + Pi/6) * n^(n - 1/2) / (2 * exp(n + 1/2 - sqrt(2*(6 + Pi^2)*n/3))). - Vaclav Kotesovec, Aug 09 2021

A362820 Number of ordered pairs of derangements on [n] that commute.

Original entry on oeis.org

1, 0, 1, 4, 33, 136, 1825, 10956, 163009, 1575568, 23894721, 280090900, 5410068961, 73066199064, 1483125027553, 25872759745756, 561027082980225, 10796395534986016, 266457543316023169, 5743345672152317988, 152031229968147150241, 3717043193920429157800, 104377807879737865769121
Offset: 0

Views

Author

Andrew Howroyd, May 05 2023

Keywords

Comments

A derangement is a permutation without fixed points. Two permutations x,y commute if x*y = y*x.

Links

Crossrefs

A053529 is the corresponding sequence for all permutations.
Cf. A000166, A362819.

Programs

  • PARI
    seq(n)=Vec(serlaplace((1 - x)^2*exp(sum(k=1, n, (x^k/k)/(1-x^k) + O(x*x^n)) + x)))

Formula

E.g.f.: (1 - x)^2 * exp(x) * B(x) where B(x) is the e.g.f. of A053529.

A274539 E.g.f.: exp(sum(bell(n)*z^n/n, n=1..infinity)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2079, 38629, 951187, 29979753, 1175837345, 56066617331, 3187704802281, 212628685506643, 16413606252207007, 1449425836362499605, 144977415195565990619, 16285937949513614300369, 2039447464767566886933057, 282862729890000953318773603
Offset: 0

Views

Author

Johannes W. Meijer, Jun 29 2016

Keywords

Comments

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
Some transform pairs, see the formula section, are: x(n) = A000027(n) and a(n) = A000262(n); x(n) = A000045(n) and a(n) = A244430(n); x(n) = A000079(n) and a(n) = A000165(n); x(n) = A000108(n) and a(n) = A213507(n); x(n) = A000142(n) and a(n) = A158876(n); x(n) = A000203(n) and a(n) = A053529(n).

Crossrefs

Cf. A036039, A000110, A000165, A000262, A053529, A158876, A213507, A244430.

Programs

  • Maple
    a := proc(n): n!*P(n) end: P := proc(n): if n=0 then 1 else P(n):= expand((1/n)*(add(x(n-k) * P(k), k=0..n-1))) fi; end: with(combinat): x := proc(n): bell(n) end: seq(a(n), n=0..18);
  • Mathematica
    nmax = 20; CoefficientList[Series[E^(Sum[BellB[n]*z^n/n, {n, 1, nmax}]), {z, 0, nmax}], z] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 29 2016 *)

Formula

a(n) = n! * P(n), with P(n) = (1/n)*(sum(x(n-k) * P(k), k=0..n-1)), n >=1 and P(0) = 1, with x(n) = A000110(n), the Bell numbers.
E.g.f.: exp(sum(x(n)*z^n/n, n=1..infinity)) with x(n) = A000110(n).

A330074 Expansion of e.g.f. Product_{k>=1} (1 - log(1 - x^k)).

Original entry on oeis.org

1, 1, 3, 14, 78, 544, 4560, 42468, 451584, 5382144, 69737760, 985265280, 15204119040, 249602065920, 4398839827200, 82834744849920, 1646970433920000, 34626184595251200, 769149445849989120, 17896198498368583680, 437123791096022016000, 11171177571932111462400
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2019

Keywords

Crossrefs

Cf. A000262, A053529, A178682.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[1 - Log[1 - x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Product[(1 + Sum[x^(i j)/i, {i, 1, nmax}]), {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: Product_{j>=1} (1 + Sum_{i>=1} x^(i*j) / i).

A330201 Expansion of e.g.f. Product_{k>=1} exp(-x^k) / (1 - x^k).

Original entry on oeis.org

1, 0, 1, 2, 21, 44, 1045, 2694, 74473, 421784, 8776521, 52518410, 1843753021, 11476952772, 387068115421, 4277646186254, 125796357803985, 1343857519264304, 53205974734877713, 621203524858308114, 25357790175078682981, 388778926109137187420
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2019

Keywords

Crossrefs

Cf. A000166, A000203, A051953, A053529, A293116, A330200.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[Exp[-x^k]/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (n - 1)! Sum[(DivisorSigma[1, k] - k) a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of A000166.
E.g.f.: exp(Sum_{k>=1} (sigma(k) / k - 1) * x^k), where sigma = A000203.
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(cototient(k)/k), where cototient = A051953.
a(0) = 1; a(n) = (n - 1)! * Sum_{k=1..n} (sigma(k) - k) * a(n-k) / (n - k)!.
a(n) = Sum_{k=0..n} binomial(n,k) * A293116(k) * A053529(n-k).
a(n) ~ sqrt(-1/Pi + Pi/6) * n^(n - 1/2) / (2 * exp(n - 1/2 - sqrt(2*(-6 + Pi^2)*n/3))). - Vaclav Kotesovec, Aug 09 2021

A086501 Number of non-commuting permutations: number of ordered pairs g, h in Symm(n) such that gh <> hg, i.e., the subgroup is non-Abelian.

Original entry on oeis.org

0, 0, 18, 456, 13560, 510480, 25326000, 1624815360, 131671008000, 13168037030400, 1593348686899200, 229442495919436800, 38775787414703539200, 7600054444782928128000, 1710012252494048735232000
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 09 2003

Keywords

Crossrefs

Cf. A000041, A053529.

Formula

a(n) = n!^2 - A053529(n) = n! * ( n! - p(n) ) where p(n) is the number of partitions of n (A000041).

Extensions

More terms from Ray Chandler, Sep 17 2003

A178882 Triangle T(n,k) = n!* A036040(n,k) read by rows, 1 <= k <= A000041(n).

Original entry on oeis.org

1, 2, 2, 6, 18, 6, 24, 96, 72, 144, 24, 120, 600, 1200, 1200, 1800, 1200, 120, 720, 4320, 10800, 7200, 10800, 43200, 10800, 14400, 32400, 10800, 720, 5040, 35280, 105840, 176400, 105840, 529200
Offset: 1

Views

Author

Alford Arnold, Jun 23 2010

Keywords

Comments

Row sums are A137341(n).

Examples

			For row n = 4 the calculations are (1 4 3 6 1) times (24 24 24 24 24 ) yielding (24 96 72 144 24)
which sums to A137341(4) = 360.
Row n has A000041(n) entries:
1;
2,2;
6,18,6;
24,96,72,144,24;
120,600,1200,1200,1800,1200,120;
720,4320,10800,7200,10800,43200,10800,14400,32400,10800,720;

Crossrefs

Cf. A000110, A053529, A137341, A036040, A178801.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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