A060435 -id:A060435 - cp's OEIS frontend

A052182 Determinant of n X n matrix whose rows are cyclic permutations of 1..n.

1, -3, 18, -160, 1875, -27216, 470596, -9437184, 215233605, -5500000000, 155624547606, -4829554409472, 163086595857367, -5952860799406080, 233543408203125000, -9799832789158199296, 437950726881001816329, -20766159817517617053696, 1041273502979112415328410

Offset: 1

Henry M. Gunn High School Mathematical Circle (Joshua Zucker), Jan 26 2000

Each row is a cyclic shift to the right by one place of the previous row. See the example below. - N. J. A. Sloane, Jan 07 2019
|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an odd number of cycles. - Vladeta Jovovic, Mar 30 2006
|a(n)| = number of functions from {1,2,...,n}->{1,2,...,n} such that of all recurrent elements the least is always mapped to the greatest. - Geoffrey Critzer, Aug 29 2013

			a(3) = 18 because this is the determinant of [(1,2,3), (3,1,2), (2,3,1) ].

T. D. Noe, Table of n, a(n) for n = 1..100
P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.

Cf. A000312, A070896, A060281, A060435.

1,seq(LinearAlgebra:-Determinant(Matrix(n,shape=Circulant[$1..n])),n=2..30); # Robert Israel, Aug 31 2014

f[n_] := Det[ Table[ RotateLeft[ Range@ n, -j], {j, 0, n - 1}]]; Array[f, 19] (* or *)
f[n_] := (-1)^(n - 1)*n^(n - 2)*(n^2 + n)/2; Array[f, 19]
(* Robert G. Wilson v, Aug 31 2014 *)
Table[Det[Table[RotateRight[Range[k],n],{n,0,k-1}]],{k,30}] (* Harvey P. Dale, Jun 20 2024 *)

(1+n)^(n-1)*binomial(n+2,n)*(-1)^(n) $ n=0..16 // Zerinvary Lajos, Apr 01 2007

a(n) = (n+1)*(-n)^(n-1)/2; \\ Altug Alkan, Dec 17 2017

a(n) = (-1)^(n-1) * n^(n-2) * (n^2 + n)/2.
E.g.f.[A052182] = E.g.f.[A000312] * E.g.f.[A000272], so A052182(unsigned) is "tree-like". E.g.f.: (T-T^2/2)/(1-T), where T=T(x) is Euler's tree function (see A000169). E.g.f. for signed sequence: (W+W^2/2)/(1+W), where W=W(x)=-T(-x) is the Lambert W function. - Len Smiley, Dec 13 2001
Conjecture: a(n) = -Res( f(n), x^n - 1), where Res is the resultant and f(n) = Sum_{k=1..n} k*x^k. - Benedict W. J. Irwin, Dec 07 2016

More terms from James Sellers, Jan 31 2000

A134095 Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).

Original entry on oeis.org

1, 0, 2, 12, 120, 1480, 22320, 396564, 8118656, 188185680, 4871980800, 139342178140, 4363291266048, 148470651659928, 5455056815237120, 215238256785814500, 9077047768435752960, 407449611073696325536, 19396232794530856894464, 976025303642559490903980

Offset: 0

Paul D. Hanna, Oct 11 2007

nonn

E.g.f. equals the square of the e.g.f. of A060435, where A060435(n) = number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

			E.g.f.: A(x) = 1 + 0*x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1480*x^5/5! + ...
The formula A(x) = 1/(1 - LambertW(-x)^2) is illustrated by:
A(x) = 1/(1 - (x + x^2 + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + ...)^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A060435; indirectly related: A062817, A132608.

Cf. A063170, A277458, A277490, A277510.

seq(simplify(GAMMA(n+1,-n)*(-exp(-1))^n),n=0..20); # Vladeta Jovovic, Oct 17 2007

CoefficientList[Series[1/(1-LambertW[-x]^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
a[x0_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x] /. x->x0
Table[a[n], {n, 0, 20}] (* Gerry Martens, May 05 2016 *)

{a(n)=sum(k=0,n,(n-k)^k*k^(n-k)*binomial(n,k))}

/* Generated by e.g.f. 1/(1 - LambertW(-x)^2 ): */
{a(n)=my(LambertW=-x*sum(k=0,n,(-x)^k*(k+1)^(k-1)/k!) +x*O(x^n)); n!*polcoeff(1/(1-subst(LambertW,x,-x)^2),n)}

a(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k).
a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*n^k/k!. - Vladeta Jovovic, Oct 17 2007
a(n) ~ n^n/2. - Vaclav Kotesovec, Nov 27 2012, simplified Nov 22 2021
a(n) = n! * [x^n] exp(n*x)/(1 + x). - Ilya Gutkovskiy, Sep 18 2018
a(n) = (-1)^n*exp(-n)*Integral_{x=-n..oo} x^n*exp(-x) dx. - Thomas Scheuerle, Jan 29 2024

A246609 Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 27, 6, 2, 0, 256, 57, 24, 6, 0, 3125, 680, 300, 120, 24, 0, 46656, 9945, 4480, 2160, 720, 120, 0, 823543, 172032, 78750, 41160, 17640, 5040, 720, 0, 16777216, 3438673, 1591296, 866460, 430080, 161280, 40320, 5040

Offset: 0

Alois P. Heinz, Aug 31 2014

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k <= n. T(0,k) = 1, T(n,k) = 0 for k > n and n > 0.

Column k > 1 is asymptotic to n^(n - 1/2 + 1/(2*k)) * sqrt(2*Pi) / (2^(1/(2*k)) * k^(1/k) * Gamma(1/(2*k))) * (1 - (3*k-1)*(k-1) * sqrt(2/n) * Gamma(1/(2*k)) / (12 * k^2 * Gamma(1/2+1/(2*k)))). - Vaclav Kotesovec, Sep 01 2014

			Triangle T(n,k) begins:
  1;
  0,      1;
  0,      4,      1;
  0,     27,      6,     2;
  0,    256,     57,    24,     6;
  0,   3125,    680,   300,   120,    24;
  0,  46656,   9945,  4480,  2160,   720,  120;
  0, 823543, 172032, 78750, 41160, 17640, 5040, 720;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000312, A060435, A246610, A246611, A246612, A246613, A246614, A246615, A246616, A246617.
Main diagonal gives A000142(n-1) for n > 0.
T(2n,n) gives A246618.

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0 or i>n, 0, add(b(n-i*j, i+k, k)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
T:= (n, k)->add(b(j, k$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(seq(T(n,k), k=0..n), n=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0 || i > n, 0, Sum[b[n-i*j, i+k, k]*(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!, {j, 0, n/i}]]]; T[0, 0] = 1; T[n_, k_] := Sum[b[j, k, k]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

E.g.f. for column k > 0: 1 / (1 - (-1)^k * LambertW(-x)^k)^(1/k). - Vaclav Kotesovec, Sep 01 2014

A070896 Determinant of the Cayley addition table of Z_{n}.

Original entry on oeis.org

0, -1, -9, 96, 1250, -19440, -352947, 7340032, 172186884, -4500000000, -129687123005, 4086546038784, 139788510734886, -5159146026151936, -204350482177734375, 8646911284551352320, 389289535005334947848, -18580248257778920521728

Offset: 1

Santi Spadaro, May 23 2002

sign

a(n) is the determinant of the n X n matrix M_(i,j) = ((i+j) mod n) where i and j range from 0 to n-1. - Benoit Cloitre, Nov 29 2002

|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an even number of cycles. E.g.f.: (1/2)*LambertW(-x)^2/(1+LambertW(-x)). - Vladeta Jovovic, Mar 30 2006

			a(3) = -9 because the determinant of {{0,1,2}, {1,2,0}, {2,0,1}} is -9.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A000312, A052182, A060281, A060435.

[(-1)^Floor(n/2)*(1/2)*(n-1)*n^(n-1): n in [1..50]]; // G. C. Greubel, Nov 14 2017

Table[(-1)^Floor[n/2]*(1/2)*(n - 1)*n^(n - 1), {n, 1, 50}] (* G. C. Greubel, Nov 14 2017 *)

a(n)=(-1)^floor(n/2)*(1/2)*(n-1)*n^(n-1)

a(n) = (-1)^floor(n/2)*(1/2)*(n-1)*n^(n-1). - Benoit Cloitre, Nov 29 2002

A116956 Number of functions f:{1,2,...,n}->{1,2,...,n} with odd cycles only.

Original entry on oeis.org

1, 1, 3, 18, 157, 1800, 25551, 432376, 8494809, 190029888, 4768313275, 132626098176, 4049755214517, 134677876657792, 4845193429684167, 187490897290080000, 7765153170076158001, 342721890859339812864, 16058392049508837366771, 796093438190851834236928

Offset: 0

Vladeta Jovovic, Mar 30 2006

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

Cf. A070896, A060281, A060435, A070896.

Cf. A212599.

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)*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[((1 + t)/(1 - t))^(1/2), {x, 0, 20}], x]  (* Geoffrey Critzer, Dec 07 2011 *)

E.g.f.: sqrt((1-LambertW(-x))/(1+LambertW(-x))).
Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) = 2*n^n, n>0. - Vladeta Jovovic, Oct 11 2007
a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(2*Pi*n^(3/4)). - Vaclav Kotesovec, Sep 24 2013

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A052182 Determinant of n X n matrix whose rows are cyclic permutations of 1..n.

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A134095 Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).

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A246609 Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A070896 Determinant of the Cayley addition table of Z_{n}.

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A116956 Number of functions f:{1,2,...,n}->{1,2,...,n} with odd cycles only.

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

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

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