A116956 -id:A116956 - cp's OEIS frontend

A273994 Number of endofunctions on [n] whose cycle lengths are Fibonacci numbers.

1, 1, 4, 27, 250, 2975, 43296, 744913, 14797036, 333393345, 8403026320, 234300271811, 7161316358616, 238108166195263, 8556626831402560, 330494399041444425, 13654219915946513296, 600870384794864432897, 28060233470995898505024, 1386000542545570348128235

Offset: 0

Alois P. Heinz, Jun 06 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000045, A060435, A116956, A273001, A273996, A273997, A273998, A305824.

b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:= $0..2;
      while f<=n do r:= r+(f-1)!*b(n-f)*
         binomial(n-1, f-1); f, g:= g, f+g
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {g, f + g}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

A273996 Number of endofunctions on [n] whose cycle lengths are factorials.

Original entry on oeis.org

1, 1, 4, 25, 218, 2451, 33952, 560407, 10750140, 235118665, 5775676496, 157448312649, 4716609543736, 154007821275595, 5443783515005760, 207093963680817511, 8436365861409555728, 366403740283162634193, 16900793597898691865920, 825115046704241167668025

Offset: 0

Alois P. Heinz, Jun 06 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000142, A060435, A116956, A272603, A273994, A273997, A273998, A305824.

b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:= $0..2;
      while f<=n do r:= r+(f-1)!*b(n-f)*
         binomial(n-1, f-1); f, g:= f*g, g+1
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {f*g, g + 1}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

A273998 Number of endofunctions on [n] whose cycle lengths are primes.

Original entry on oeis.org

1, 0, 1, 8, 75, 904, 13255, 229536, 4587961, 103971680, 2634212961, 73787255200, 2264440519891, 75563445303072, 2724356214102055, 105546202276277504, 4373078169296869425, 192970687573630633216, 9035613818754820178689, 447469496697658409400960

Offset: 0

Alois P. Heinz, Jun 06 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000040, A060435, A116956, A218002, A273994, A273996, A273997, A305824.

b:= proc(n) option remember; local r, p;
      if n=0 then 1 else r, p:=0, 2;
      while p<=n do r:= r+(p-1)!*b(n-p)*
         binomial(n-1, p-1); p:= nextprime(p)
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

b[n_] := b[n] = Module[{r, p}, If[n == 0, 1, {r, p} = {0, 2}; While[p <= n, r = r + (p - 1)!*b[n - p]*Binomial[n-1, p-1]; p = NextPrime[p]]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

A305824 Number of endofunctions on [n] whose cycle lengths are triangular numbers.

Original entry on oeis.org

1, 1, 3, 18, 157, 1776, 24807, 413344, 8004537, 176630400, 4374300331, 120136735104, 3623854678677, 119102912981248, 4236492477409935, 162152320065532416, 6645233337842716273, 290321208589666369536, 13469914225467040015827, 661442143465113960448000

Offset: 0

Alois P. Heinz, Jun 10 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000217, A060435, A116956, A193374, A205799, A273994, A273996, A273998.

b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:=$0..2;
      while f<=n do r, f, g:= r+(f-1)!*
         b(n-f)*binomial(n-1, f-1), f+g, g+1
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, {r, f, g} = {r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1], f + g, g + 1}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 15 2018, after Alois P. Heinz *)

A273997 Number of endofunctions on [n] whose cycle lengths are squares.

Original entry on oeis.org

1, 1, 3, 16, 131, 1446, 19957, 329344, 6315129, 137942380, 3382214291, 92014156224, 2751300514987, 89701699067176, 3167429783609925, 120428877629249536, 4905431165356442993, 213120603686615692176, 9837426739843075654819, 480775495859934668704000

Offset: 0

Alois P. Heinz, Jun 06 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000290, A060435, A116956, A205801, A273994, A273996, A273998.

b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:=0, 1, 3;
      while f<=n do r:= r+(f-1)!*b(n-f)*
         binomial(n-1, f-1); f, g:= f+g, g+2
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 3}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {f + g, g + 2}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

A212599 Number of functions on n labeled points to themselves (endofunctions) such that the number of cycles of f that have each even size is even.

Original entry on oeis.org

1, 1, 3, 18, 160, 1875, 27126, 466186, 9275064, 209654325, 5307031000, 148720701426, 4570816040352, 152874605142727, 5527634477245440, 214862754390554250, 8934811701563214976, 395788795274021394729, 18606559519007667893376, 925222631836457779380370, 48518852386696450625510400

Offset: 0

Geoffrey Critzer, May 22 2012

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A003483, A246951, A116956.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 2))<>0, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> add(b(j, j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 08 2014

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];p=Product[Cosh[t^(2i)/(2i)],{i,1,nn}];Range[0,nn]! CoefficientList[Series[((1+t)/(1-t))^(1/2) p,{x,0,nn}],x]

E.g.f.: ((1+T(x))/(1-T(x)))^(1/2) * Product_{i>=1} cosh(T(x)^(2*i)/(2*i)) where T(x) is the e.g.f. for A000169.

Maple program fixed by Vaclav Kotesovec, Sep 13 2014

A202013 The number of functions f:{1,2,...,n}->{1,2,...,n} that have an odd number of odd length cycles and no even length cycles.

Original entry on oeis.org

0, 1, 2, 12, 100, 1120, 15606, 260344, 5056136, 112026240, 2788230250, 77009739136, 2337124786668, 77302709780608, 2767629599791070, 106631592312384000, 4398877912885363216, 193450993635808976896, 9034380526387410161874, 446519425974262943518720, 23284829853408862172112500

Offset: 0

Geoffrey Critzer, Dec 08 2011

nonn

The number of endofunctions with an odd number of recurrent elements.

It appears that almost all endofunctions have an even number of recurrent elements.

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A060435, A116956.

b:= proc(n, t) option remember; `if`(n=0, t, add(
      `if`(j::odd, (j-1)!*b(n-j, 1-t)*
       binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> add(b(j, 0)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 20 2016

t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[Sinh[Log[((1 + t)/(1 - t))^(1/2)]], {x, 0, 20}], x]
CoefficientList[Series[(((1-LambertW[-x])/(1+LambertW[-x]))^(1/2))/2 - 1/(2*((1-LambertW[-x])/(1+LambertW[-x]))^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 24 2013 *)

E.g.f.: sinh(log(((1-LambertW(-x))/(1+LambertW(-x)))^(1/2))). - corrected by Vaclav Kotesovec, Sep 24 2013

a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(4*Pi*n^(3/4)) * (1+7*Pi/(24*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Sep 24 2013

A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

Original entry on oeis.org

1, 1, 3, 12, 73, 580, 5601, 63994, 844929, 12647016, 211616065, 3914510446, 79320037281, 1747219469164, 41569414869633, 1062343684252530, 29023112392093441, 844101839207139280, 26038508978625589377, 849150487829425227094, 29189561873274715264545

Offset: 0

Geoffrey Critzer, Jul 25 2016

nonn

Equivalently, these are the functions counted by A116956 with the additional constraint that every element is mapped to a recurrent element. A recurrent element is an element on a cycle in the functional digraph.

Alois P. Heinz, Table of n, a(n) for n = 0..411

Cf. A000246, A006153, A009444, A116956, A216401.

b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
       (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> add(b(j)*j^(n-j)*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

nn = 20; Range[0, nn]! CoefficientList[Series[Sqrt[(1 + z*Exp[z])/(1 - z*Exp[z])], {z, 0, nn}], z]

default(seriesprecision, 30);
S=sqrt((1 + x*exp(x))/(1 - x*exp(x)));
v=Vec(S); for(n=2,#v-1,v[n+1]*=n!); v \\ Charles R Greathouse IV, Jul 29 2016

E.g.f.: sqrt((1 + z*exp(z))/(1 - z*exp(z))).
Exponential transform of A216401.
a(n) ~ 2 * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^n * exp(n)). - Vaclav Kotesovec, Jun 26 2022

cp's OEIS Frontend

A273994 Number of endofunctions on [n] whose cycle lengths are Fibonacci numbers.

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Maple

Mathematica

A273996 Number of endofunctions on [n] whose cycle lengths are factorials.

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Maple

Mathematica

A273998 Number of endofunctions on [n] whose cycle lengths are primes.

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Maple

Mathematica

A305824 Number of endofunctions on [n] whose cycle lengths are triangular numbers.

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Maple

Mathematica

A273997 Number of endofunctions on [n] whose cycle lengths are squares.

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Maple

Mathematica

A212599 Number of functions on n labeled points to themselves (endofunctions) such that the number of cycles of f that have each even size is even.

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Maple

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A202013 The number of functions f:{1,2,...,n}->{1,2,...,n} that have an odd number of odd length cycles and no even length cycles.

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Programs

Maple

Mathematica

Formula

A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

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Maple

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