A285849 -id:A285849 - cp's OEIS frontend

A104150 Shifted factorial numbers: a(0)=0, a(n) = (n-1)!.

0, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 0

Miklos Kristof, Mar 08 2005

E.g.f.: Sum_{n>=1} (n-1)!*x^n/n! = Sum_{n>=1} x^n/n.
The shift law of the e.g.f.: if Sum_{n>=0} a(n)*x^n/n! = f(x), then Sum_{n>=0} a(n+1)*x^n/n! = d/dx f(x) and Sum_{n>=1} a(n-1)*x^n/n! = Integral f(x) dx.
The e.g.f. of A000142 (= n!) is 1/(1-x), so the e.g.f. of a(n)=(n-1)! is integral 1/(1-x) = -log(1-x).

A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p.141 (10.19)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. N. Gladkovskii, Analysis Of The Continued Fractions (in Russian), see p.79 (5.1.21)

Cf. A000142.
Column k=1 of A285849.
Main diagonal of A295027 (for n > 0).

[0] cat [Factorial(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 25 2012

Join[{0,1},Range[20]!] (* Harvey P. Dale, Dec 09 2013 *)

my(x='x+O('x^30)); concat([0], Vec(serlaplace(-log(1-x)))) \\ G. C. Greubel, May 15 2018

[stirling_number1(n,1) for n in range(0, 22)] # Zerinvary Lajos, May 16 2009

E.g.f. -log(1-x) = x + x^2/2 + x^3/3 + ... + x^n/n + ...
G.f.: x+x^2/(G(0)-x) where G(k) = 1 - (k+1)*x/(1 - x*(k+2)/G(k+1)); G(0) = W(1,1;-x)/W(1,2;-x), W(a,b,x) = 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)], x-> -x;  (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 14 2012
E.g.f.: (-x + 5*x^2/2 - 11*x^3/6 + x^4/4 + x^5/(W(0)-x)/4)/(x-1)^3 where W(k)= (x + 1)*k + x + 5 - x*(k+2)*(k+5)/W(k+1); see [S. N. Gladkovskii, p. 79 (5.1.21)]; (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 15 2012
G.f.: A(x) = Integral_{t>=0} x*exp(-t)/(1-x*t) dt = x/G(0) where G(k) = 1 - x*(k+1)/(1 - x*(k+1)/G(k+1)); (continued fraction due to L. Euler and E. N. Laguerre). - Sergei N. Gladkovskii, Dec 24 2012
G.f.: x + x/Q(0), where Q(k)= 1/x - (2*k+2) - (k+2)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013
G.f.: x/Q(0), where Q(k) = 1 - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
G.f.: x*G(0), where G(k) = 1 + x*(2*k+1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
G.f.: x*G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 07 2013

A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 6, 1, 0, 1, 11, 18, 1, 0, 1, 30, 75, 40, 1, 0, 1, 52, 420, 350, 75, 1, 0, 1, 126, 1218, 3080, 1225, 126, 1, 0, 1, 219, 4242, 17129, 15750, 3486, 196, 1, 0, 1, 510, 14563, 82488, 152355, 63756, 8526, 288, 1, 0, 1, 896, 42930, 464650, 1049895, 954387, 217560, 18600, 405, 1

Offset: 0

Alois P. Heinz, Apr 27 2017

			T(3,1) = 1: 123.
T(3,2) = 6: 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
T(3,3) = 1: 1|2|3.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   6,    1;
  0, 1,  11,   18,     1;
  0, 1,  30,   75,    40,     1;
  0, 1,  52,  420,   350,    75,    1;
  0, 1, 126, 1218,  3080,  1225,  126,   1;
  0, 1, 219, 4242, 17129, 15750, 3486, 196, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A057427, A285917, A285918, A285919, A285920, A285921, A285922, A285923, A285924, A285925.
Main diagonal and first lower diagonal give: A000012, A002411.
Row sums give A120774.
T(2n,n) gives A285926.
Cf. A048993, A131689, A226874, A285849.

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*x^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n-i*j, i-1, p+j]*x^j*multinomial[n, Join[{n-i*j}, Table[i, j]]]/ j!^2, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)

A285862 Number of permutations of [2n] with n ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 1, 19, 1005, 62601, 6061545, 868380535, 142349568361, 27564092244689, 6325532235438273, 1673378033771898675, 505141951803309946125, 170002056228253072537065, 63255335047795174479833625, 25805276337820748477042392695, 11427131417576257617280878155625

Offset: 0

Alois P. Heinz, Apr 27 2017

nonn

			a(1) = 1: (12).
a(2) = 19: (123)(4), (4)(123), (132)(4), (4)(132), (124)(3), (3)(124), (142)(3), (3)(142), (134)(2), (2)(134), (143)(2), (2)(143), (1)(234), (234)(1), (1)(243), (243)(1),  (12)(34), (13)(24), (14)(23).

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

Cf. A187646, A285849, A285926.

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
    end:
a:= n-> coeff(b(2*n$2, 0), x, n):
seq(a(n), n=0..20);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];
a[n_] := Coefficient[b[2n, 2n, 0], x, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from Maple *)

a(n) = A285849(2n,n).

A196301 The number of ways to linearly order the cycles in each permutation of {1,2,...,n} where two cycles are considered identical if they have the same length.

Original entry on oeis.org

1, 1, 2, 9, 44, 270, 2139, 18837, 186808, 2070828, 25861140, 350000640, 5145279611, 81492295079, 1381583542234, 25097285838765, 484602684624080, 9894705390149400, 213418984780492164, 4842425874827849868, 115231446547162291200, 2874808892527026177240

Offset: 0

Geoffrey Critzer, Sep 30 2011

nonn

			a(4) = 44 because in the conjugacy classes of S(4): (4), (3)(1), (2)(2), (2)(1)(1), (1)(1)(1)(1) there are (respectively) 6 permutations times 1 arrangement, 8 permutations times 2 arrangements, 3 permutations times 1 arrangement, 6 permutations times 3 arrangements, and 1 permutation times 1 arrangement.  So 6*1+8*2+3*1+6*3+1*1 = 44.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A120774.

Row sums of A285849.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      (p+n)!/n!, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 27 2017

Needs["Combinatorica`"]; f[{x_, y_}]:= x^y y!; Table[Total[Table[n!, {PartitionsP[n]}]/Apply[Times, Map[f, Map[Tally, Partitions[n]], {2}], 2] * Apply[Multinomial, Map[Last, Map[Tally, Partitions[n]], {2}], 2]], {n, 0, 20}]

A285853 Number of permutations of [n] with two ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 6, 19, 100, 508, 3528, 24876, 219168, 1980576, 21257280, 234434880, 2972885760, 38715943680, 566931294720, 8514866707200, 141468564787200, 2407290355814400, 44753976117043200, 850965783594393600, 17505896073523200000, 367844990453821440000

Offset: 2

Alois P. Heinz, Apr 27 2017

nonn

			a(2) = 1: (1)(2).
a(3) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
a(4) = 19: (123)(4), (4)(123), (132)(4), (4)(132), (124)(3), (3)(124), (142)(3), (3)(142), (134)(2), (2)(134), (143)(2), (2)(143), (1)(234), (234)(1), (1)(243), (243)(1),  (12)(34), (13)(24), (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 2..450
Wikipedia, Permutation

Column k=2 of A285849.

Cf. A285917.

a:= n-> 2*add(binomial(n, k)*(k-1)!*(n-k-1)!, k=1..n/2)-
        `if`(n::even, 3/2*binomial(n, n/2)*(n/2-1)!^2, 0):
seq(a(n), n=2..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 19][n],
     ((2*n-1)*(n-1)*a(n-1)+(n-2)*(2*n^2-5*n-1)*a(n-2)
      -(n-3)^2*((2*n^2-5*n+4)*a(n-3)+(n-4)^2*a(n-4)))/(2*n))
    end:
seq(a(n), n=2..25);

Table[(n-1)!*(2*HarmonicNumber[n] - (3 + (-1)^n)/n), {n, 2, 25}] (* Vaclav Kotesovec, Apr 29 2017 *)

A285854 Number of permutations of [n] with three ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 18, 105, 1005, 6762, 61572, 558548, 5807700, 62757288, 777291768, 9831740256, 139111566048, 2048834965824, 32758018496640, 545051532176640, 9812211976039680, 182219827628874240, 3627461543458659840, 74765368810365696000, 1632210845693218560000

Offset: 3

Alois P. Heinz, Apr 27 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..450
Wikipedia, Permutation

Column k=3 of A285849.

Cf. A285918.

b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 4)
    end:
a:= n-> coeff(b(n$2, 0), x, 3):
seq(a(n), n=3..25);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]], {x, 0, 4}];
a[n_] := Coefficient[b[n, n, 0], x, 3];
Table[a[n], {n, 3, 25}] (* Jean-François Alcover, May 30 2018, from Maple *)

A285855 Number of permutations of [n] with four ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 40, 430, 6300, 62601, 706608, 8985560, 107911760, 1439518696, 20364348576, 304923257184, 4772610024000, 80570363703696, 1409795519233536, 26263500315144192, 511153327733815296, 10464902116976779776, 223154458395064842240, 5010190272214829475840

Offset: 4

Alois P. Heinz, Apr 27 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..450
Wikipedia, Permutation

Column k=4 of A285849.

Cf. A285919.

b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 5)
    end:
a:= n-> coeff(b(n$2, 0), x, 4):
seq(a(n), n=4..25);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]], {x, 0, 4}];
a[n_] := Coefficient[b[n, n, 0], x, 4];
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, May 30 2018, from Maple *)

A285856 Number of permutations of [n] with five ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 75, 1400, 28700, 431445, 6061545, 101969450, 1511771250, 24207257360, 398261143280, 7126152051200, 125804137586400, 2407653784466640, 47180921713154640, 978889889736934560, 20912829727776156000, 470414513170923511680, 10891943467262968682880

Offset: 5

Alois P. Heinz, Apr 27 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..450
Wikipedia, Permutation

Column k=5 of A285849.

Cf. A285920.

b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 6)
    end:
a:= n-> coeff(b(n$2, 0), x, 5):
seq(a(n), n=5..25);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]], {x, 0, 6}];
a[n_] := Coefficient[b[n, n, 0], x, 5];
Table[a[n], {n, 5, 25}] (* Jean-François Alcover, May 30 2018, from Maple *)

A285857 Number of permutations of [n] with six ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 126, 3822, 105336, 2312163, 41420610, 868380535, 16453085220, 312866654100, 6063351173880, 127050688947000, 2603853165950400, 56141875342402480, 1242418296237553440, 28627706535786406800, 683460419058369489600, 16802904218347937067840

Offset: 6

Alois P. Heinz, Apr 27 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..450
Wikipedia, Permutation

Column k=6 of A285849.

Cf. A285921.

b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 7)
    end:
a:= n-> coeff(b(n$2, 0), x, 6):
seq(a(n), n=6..25);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]], {x, 0, 7}];
a[n_] := Coefficient[b[n, n, 0], x, 6];
Table[a[n], {n, 6, 25}] (* Jean-François Alcover, May 30 2018, from Maple *)

A285858 Number of permutations of [n] with seven ordered cycles such that equal-sized cycles are ordered with increasing least elements.

Original entry on oeis.org

1, 196, 9114, 330750, 10094931, 234138366, 5932023097, 142349568361, 3233779086538, 74147737383720, 1785843031638120, 42966579274786440, 1047584220405271360, 26222209747260881200, 671966452779878874800, 17944599541172975286000, 485789620369911667323360

Offset: 7

Alois P. Heinz, Apr 27 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..450
Wikipedia, Permutation

Column k=7 of A285849.

Cf. A285922.

b:= proc(n, i, p) option remember; series(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)), x, 8)
    end:
a:= n-> coeff(b(n$2, 0), x, 7):
seq(a(n), n=7..25);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Series[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]], {x, 0, 8}];
a[n_] := Coefficient[b[n, n, 0], x, 7];
Table[a[n], {n, 7, 25}] (* Jean-François Alcover, May 30 2018, from Maple *)

cp's OEIS Frontend

A104150 Shifted factorial numbers: a(0)=0, a(n) = (n-1)!.

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A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A285862 Number of permutations of [2n] with n ordered cycles such that equal-sized cycles are ordered with increasing least elements.

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Maple

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A196301 The number of ways to linearly order the cycles in each permutation of {1,2,...,n} where two cycles are considered identical if they have the same length.

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Maple

Mathematica

A285853 Number of permutations of [n] with two ordered cycles such that equal-sized cycles are ordered with increasing least elements.

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Maple

Mathematica

A285854 Number of permutations of [n] with three ordered cycles such that equal-sized cycles are ordered with increasing least elements.

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Maple

Mathematica

A285855 Number of permutations of [n] with four ordered cycles such that equal-sized cycles are ordered with increasing least elements.

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Maple

Mathematica

A285856 Number of permutations of [n] with five ordered cycles such that equal-sized cycles are ordered with increasing least elements.

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