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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A239749 Number of ordered pairs of functions f,g on a set of n elements into itself satisfying f(f(x)) = g(f(g(x))).

Original entry on oeis.org

1, 1, 6, 87, 2056, 69605, 3201696, 190933435
Offset: 0

Views

Author

Chad Brewbaker, Mar 26 2014

Keywords

Examples

			a(0) = a(1) = 1 since there is only one endofunction for n=0 or 1 and the equation is satisfied trivially. For n=2, each endofunction f on {1,2} is represented by [f(1),f(2)]. The list of a(2) = 6 pairs (f,g) which satisfy the equation is ([1,1], [1,1]), ([1,1], [1,2]), ([1,2], [1,2]), ([1,2], [2,1]), ([2,2], [1,2]), ([2,2], [2,2]). - _Michael Somos_, Mar 26 2014

Crossrefs

Cf. A000248, A000949.
Related sequences: A053529, A181162, A239749-A239785, A239836-A239841.

Extensions

a(6)-a(7) from Giovanni Resta, Mar 26 2014

A346548 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(-x).

Original entry on oeis.org

1, 1, 2, 6, 42, 175, 2015, 10843, 157388, 1240377, 20118077, 172029231, 4052166250, 36360150385, 952965601471, 11194257455977, 316421367496344, 3722989943371217, 134504815853036649, 1641201826969536379, 67298415781492985366, 935342610632498431241, 40176825083871581430723
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002743.
The first negative term is a(71) = -1.2234788... * 10^104.

Crossrefs

Cf. A000203, A002743, A053529, A346545, A346546, A346547.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Exp[-x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002743[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002743[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp( exp(-x) * Sum_{k>=1} sigma(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002743(k) * a(n-k).

A346841 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^sin(x).

Original entry on oeis.org

1, 0, 2, 9, 40, 360, 2480, 28833, 266936, 3562920, 45634258, 659631225, 10231705196, 176661237948, 3080315922294, 59430009554685, 1217593208993232, 25766943601055184, 583245289316927058, 13861911731632256457, 343615639889119016556, 8925102256331257339140, 242399591002192962709230
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000203, A053529, A335626, A346547, A346941, A347774.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^sin(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, sigma(k)*x^k/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, x^k/(k*(1-x^k))))))

Formula

E.g.f.: exp( sin(x) * Sum_{k>=1} sigma(k)*x^k/k ).
E.g.f.: exp( sin(x) * Sum_{k>=1} x^k/(k*(1 - x^k)) ).

A362819 Number of ordered pairs of involutions on [n] that commute.

Original entry on oeis.org

1, 1, 4, 10, 52, 196, 1216, 5944, 42400, 250912, 2008576, 13815616, 122074624, 950640640, 9158267392, 79258479616, 824644235776, 7823203807744, 87245790791680, 897748312609792, 10665239974537216, 118040852776093696, 1486172381689544704, 17572063073426206720, 233446797379437248512
Offset: 0

Views

Author

Andrew Howroyd, May 05 2023

Keywords

Comments

Two involutions x,y on [n] commute if x*y = y*x (i.e. x(y(i)) = y(x(i)) for i in [n]).

Links

Crossrefs

Column k=2 of A362824.
A053529 is the corresponding sequence for all permutations.
Cf. A000085, A000898, A181162, A362820, A362825.

Programs

  • PARI
    b(n,f) = {sum(k=0, n\2, f(k)*binomial(n,2*k)*(2*k)!/(k!*2^k))}
a(n) = {b(n, k->b(n-2*k, j->1)*b(k, j->2^(k-j)))}
  • PARI
    seq(n)=Vec(serlaplace(exp(x + 3*x^2/2 + x^4/4 + O(x*x^n))))

Formula

a(n) = Sum_{k=0..floor(n/2)} A000085(n-2*k) * A000898(k) * binomial(n,2*k) * (2*k)! / (k!*2^k).
E.g.f.: exp(x + 3*x^2/2 + x^4/4).

A293301 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} 1/(Sum_{j=0..k} (-1)^j*x^(j*i)/j!).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 1, 3, 18, 0, 1, 1, 3, 12, 120, 0, 1, 1, 3, 13, 66, 840, 0, 1, 1, 3, 13, 74, 450, 7920, 0, 1, 1, 3, 13, 73, 510, 3510, 75600, 0, 1, 1, 3, 13, 73, 500, 4130, 32760, 887040, 0, 1, 1, 3, 13, 73, 501, 4040, 38430, 335160, 10886400, 0, 1, 1, 3
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2017

Keywords

Examples

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   4,   3,   3,   3, ...
   0,  18,  12,  13,  13, ...
   0, 120,  66,  74,  73, ...
   0, 840, 450, 510, 500, ...

Links

Crossrefs

Columns k=0..2 give A000007, A053529, A293302.
Rows n=0 gives A000012.
Main diagonal gives A000262.
Cf. A293135, A293139, A293299.

A354623 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^x.

Original entry on oeis.org

1, 0, 2, 9, 44, 390, 2754, 32760, 310064, 4244184, 54887400, 818615160, 12909921672, 225872515440, 4045885572624, 79360837887240, 1649832369335040, 35666417240193600, 822291935260976064, 19830352438530840960, 501144432316767688320, 13229590606682042436480
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A355064, A356554.
Cf. A000203, A053529, A066166, A356335, A356565.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k)^x))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * sigma(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A073584 Commuting even permutations: number of ordered pairs g, h in the alternating group A_n such that gh = hg.

Original entry on oeis.org

1, 1, 9, 48, 300, 2520, 22680, 282240, 3265920, 43545600, 618710400, 10298534400, 171243072000, 3138418483200, 61460695296000, 1286751578112000, 27743619391488000, 640237370572800000, 15448927751921664000, 394130125324615680000, 10422552203028725760000, 288306186674956369920000
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003

Keywords

Links

Crossrefs

Cf. A000702, A053529.

Formula

For n>=2 : a(n) = (n!/2) * A000702(n).

Extensions

More terms from N. J. A. Sloane (based on A000702), Dec 31 2006

A178801 Write n! partition(n) times.

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 24, 24, 24, 24, 24, 120, 120, 120, 120, 120, 120, 120, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 40320, 40320, 40320, 40320, 40320, 40320
Offset: 0

Views

Author

Alford Arnold, Jun 15 2010

Keywords

Examples

			for n = 0, 1, 2, 3, ... partition(n) (A000041) begins 1,1,2,3,5,7,11,15,... therefore Triangle begins:
   1;
   1;
   2,  2;
   6,  6,  6;
  24, 24, 24, 24, 24;
  ...

Crossrefs

Cf. A000041 (shape sequence), A000142, A053529 (row sums), A066186, A036042 (write n partition(n) times).

Programs

  • Mathematica
    Table[PadRight[{},PartitionsP[n],n!],{n,10}]//Flatten (* Harvey P. Dale, Apr 05 2020 *)

Extensions

More terms from R. J. Mathar, Jun 19 2010
Name clarified by N. J. A. Sloane, Apr 05 2020

A239837 Number of ordered pairs of permutation functions on n elements satisfying f(f(x)) = g(f(g(x))).

Original entry on oeis.org

1, 1, 2, 12, 96, 600, 6480, 85680, 1048320, 16692480, 315705600, 5468601600, 117834393600, 2951607859200, 68958028339200, 1856897602560000, 58228124258304000, 1721171464556544000, 56379302852640768000, 2111880588197732352000, 76033053559536353280000
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(20) from Hiroaki Yamanouchi, Mar 12 2015

A239838 Number of ordered pairs of permutation functions on n elements where f(f(f(x))) = g(g(g(x))).

Original entry on oeis.org

1, 1, 2, 12, 96, 600, 9360, 146160, 1935360, 41368320, 1092268800, 23111827200, 700300339200, 26141033318400, 810322216704000, 31624796915712000, 1537594906079232000, 63980698878480384000, 3128302230597992448000, 187447679378885173248000
Offset: 0

Views

Author

Chad Brewbaker, Mar 27 2014

Keywords

Links

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.
a(n) = n! * A232207(n) for n>0. - Alois P. Heinz, Jul 23 2014

Extensions

a(8)-a(12) from Giovanni Resta, Mar 27 2014
a(13) from Alois P. Heinz, Jul 23 2014
a(14)-a(19) from Hiroaki Yamanouchi, Mar 12 2015
Previous Showing 31-40 of 53 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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