A091137 -id:A091137 - cp's OEIS frontend

A232193 Numerators of the expected value of the length of a random cycle in a random n-permutation.

1, 3, 23, 55, 1901, 4277, 198721, 16083, 14097247, 4325321, 2132509567, 4527766399, 13064406523627, 905730205, 13325653738373, 362555126427073, 14845854129333883, 57424625956493833, 333374427829017307697, 922050973293317, 236387355420350878139797

Offset: 1

Geoffrey Critzer, Nov 20 2013

In this experiment we randomly select (uniform distribution) an n-permutation and then randomly select one of the cycles from that permutation. Cf. A102928/A175441 which gives the expected cycle length when we simply randomly select a cycle.

			Expectations for n=1,... are 1/1, 3/2, 23/12, 55/24, 1901/720, 4277/1440, 198721/60480, 16083/4480, ... = A232193/A232248
For n=3 there are 6 permutations.  We have probability 1/6 of selecting (1)(2)(3) and the cycle size is 1.  We have probability 3/6 of selecting a permutation with cycle type (1)(23) and (on average) the cycle length is 3/2.  We have probability 2/6 of selecting a permutation of the form (123) and the cycle size is 3.  1/6*1 + 3/6*3/2 + 2/6*3 = 23/12.

Alois P. Heinz, Table of n, a(n) for n = 1..250

Denominators are A232248.
Cf. A028417(n)/n! the expected value of the length of the shortest cycle in a random n-permutation.
Cf. A028418(n)/n! the expected value of the length of the longest cycle in a random n-permutation.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j
      *b(n-i*j, i-1) *x^j, j=0..n/i))))
    end:
a:= n->numer((p->add(coeff(p, x, i)/i, i=1..n))(b(n$2))/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 21 2013
# second Maple program:
a:= n-> numer(add(abs(combinat[stirling1](n, i))/i, i=1..n)/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 23 2013

Table[Numerator[Total[Map[Total[#]!/Product[#[[i]],{i,1,Length[#]}]/Apply[Times,Table[Count[#,k]!,{k,1,Max[#]}]]/(Total[#]-1)!/Length[#]&,Partitions[n]]]],{n,1,25}]

a(n) = Numerator( 1/(n-1)! * Sum_{i=1..n} A132393(n,i)/i ). - Alois P. Heinz, Nov 23 2013

a(n) = numerator(Sum_{k=0..n} A002657(k)/A091137(k)) (conjectured). - Michel Marcus, Jul 19 2019

A232248 Denominators of the expected length of a random cycle in a random permutation.

Original entry on oeis.org

1, 2, 12, 24, 720, 1440, 60480, 4480, 3628800, 1036800, 479001600, 958003200, 2615348736000, 172204032, 2414168064000, 62768369664000, 2462451425280000, 9146248151040000, 51090942171709440000, 136216903680000, 33720021833328230400000, 67440043666656460800000

Offset: 1

Geoffrey Critzer, Nov 21 2013

Alois P. Heinz, Table of n, a(n) for n = 1..250

Numerators are A232193.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j
      *b(n-i*j, i-1) *x^j, j=0..n/i))))
    end:
a:= n->denom((p->add(coeff(p, x, i)/i, i=1..n))(b(n$2))/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 21 2013
# second Maple program:
a:= n-> denom(add(abs(combinat[stirling1](n, i))/i, i=1..n)/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 23 2013

Table[Denominator[Total[Map[Total[#]!/Product[#[[i]],{i,1,Length[#]}]/Apply[Times,Table[Count[#,k]!,{k,1,Max[#]}]]/(Total[#]-1)!/Length[#]&,Partitions[n]]]],{n,1,25}]

a(n) = Denominator( 1/(n-1)! * Sum_{i=1..n} A132393(n,i)/i ). - Alois P. Heinz, Nov 23 2013

a(n) = denominator(Sum_{k=0..n} A002657(k)/A091137(k)) (conjectured). - Michel Marcus, Jul 19 2019

A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 1, 1, 1, 41, 181, 1, 5, 1, 1, 109, 97, 41, 35, 1, 1, 1, 853, 551, 173, 107, 1, 7, 1, 1, 19, 13579, 1313, 307, 203, 7, 1, 1, 1, 1679, 251, 1081, 5969, 1681, 1169, 5, 3, 1, 1, 1537, 3169, 4913, 13583, 3481, 7819, 101, 11, 5, 1, 1, 18167

Offset: 1

Tilman Neumann, Oct 03 2008, Oct 04 2008

This triangle T[n,k] is given by the numerators of rational coefficients R[n,k] appearing in a certain series expansion of 1/S(x) around x0=0,
where S(x) = - x*log(x) - (1-x)*log(1-x) is the Bernoulli trial entropy.
The series is
1/S(x) = 1/(x*(1-log(x))) + sum_{n=1..inf} x^(n-1) * sum_{k=1..n} R[n,k]/(1-log(x))^(k+1)
= 1/(x*(1-log(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-log(x))^k)
The first rationals R[n,k] are
1/2
1/6 1/4
1/12 1/6 1/8
1/20 1/9 1/8 1/16
1/30 7/90 5/48 1/12 1/32
1/42 41/720 181/2160 1/12 5/96 1/64
1/56 109/2520 97/1440 41/540 35/576 1/32 1/128
See A145177 for the denominators of R[n,k] and A145178 for numerators scaled to denominators given by A091137.

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened).

Cf. A145177, A145178, A091137.

f:= -x*log(x)-(1-x)*log(1-x):
S:= map(normal,eval(series(x*(1-ln(x))/f, x, 12),ln(x)=1-1/t)):
for n from 1 to 141 do
  C:= coeff(S,x,n);
  for k from 1 to n do T[n,k]:= numer(coeff(C,t,k));
 od
od:
seq(seq(T[n,k],k=1..n),n=1..10); # Robert Israel, Jul 09 2015

ORDER:=14: expand(_invert(series(-x*ln(x)-(1-x)*ln(1-x), x=0)));

A145177 Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy).

Original entry on oeis.org

2, 6, 4, 12, 6, 8, 20, 9, 8, 16, 30, 90, 48, 12, 32, 42, 720, 2160, 12, 96, 64, 56, 2520, 1440, 540, 576, 32, 128, 72, 25200, 10080, 2592, 1728, 24, 384, 256, 90, 700, 302400, 22680, 5184, 4320, 256, 96, 512, 110, 75600, 6720, 21600, 108864, 34560, 34560, 288

Offset: 1

Tilman Neumann, Oct 03 2008, Oct 04 2008

This triangle T[n,k] is given by the denominators of rational coefficients R[n,k] appearing in a certain series expansion of 1/S(x) around x=0,
where S(x) = -x*log(x) - (1-x)*log(1-x) is the Bernoulli trial entropy.
The series is
1/S(x) = 1/(x*(1-log(x))) + sum_{n=1..inf} x^(n-1) * sum_{k=1..n} R[n,k]/(1-log(x))^(k+1)
= 1/(x*(1-log(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-log(x))^k).
The first rationals R[n,k] are
1/2
1/6     1/4
1/12    1/6       1/8
1/20    1/9       1/8      1/16
1/30    7/90      5/48     1/12    1/32
1/42   41/720   181/2160   1/12    5/96   1/64
1/56  109/2520   97/1440  41/540  35/576  1/32  1/128
The LCM of the rows of T[n,k], i.e., A003418(A145177(n,1), ..., A145177(n,n)), is just A091137(n).
See A145176 for the numerators of R[n,k] and A145178 for the numerators scaled to denominators A091137.

Robert Israel, Table of n, a(n) for n = 1..10011(first 141 rows, flattened)

Cf. A003418, A091137, A145176, A145178.

f:= -x*log(x)-(1-x)*log(1-x):
S:= map(normal,eval(series(x*(1-ln(x))/f, x, 12),ln(x)=1-1/t)):
for n from 1 to 10 do
  C:= coeff(S,x,n);
  for k from 1 to n do T[n,k]:= denom(coeff(C,t,k)) od
od:
seq(seq(T[n,k],k=1..n),n=1..10); # Robert Israel, Jul 09 2015

ORDER:=14: expand(_invert(series(-x*ln(x)-(1-x)*ln(1-x), x=0)));

A165281 a(n) = (n+1)(6n^4 - 51n^3 + 161n^2 - 251*n + 251).

Original entry on oeis.org

251, 232, 243, 224, 475, 2376, 9107, 26368, 63099, 132200, 251251, 443232, 737243, 1169224, 1782675, 2629376, 3770107, 5275368, 7226099, 9714400, 12844251, 16732232, 21508243, 27316224, 34314875, 42678376, 52597107, 64278368, 77947099

Offset: 0

Paul Curtz, Sep 13 2009

The sequence is the numerators of the fifth column of the array on page 56 of the reference. The denominators are A091137(4)=720.
The sequence is the binomial transform of the quasi-finite 251, -19, 30, -60, 360, 720, 0, 0, 0, 0, ...
The fifth differences are (constant) 720; the fourth differences are 720*n + 360.

P. Curtz, Integration numerique des systemes differentiels a conditions initiales, C.C.S.A., Arcueil, 1969.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A157371, A152064.

[(n+1)*(6*n^4-51*n^3+161*n^2-251*n+251): n in [0..30]]; // Vincenzo Librandi, Aug 07 2011

Table[(n+1)(6n^4-51n^3+161n^2-251n+251),{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{251,232,243,224,475,2376},30] (* Harvey P. Dale, Aug 20 2014 *)

a(n) mod 10 = A010879(n+1).
a(n+1) - a(n) = A157411(n).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: ( 251 - 1274*x + 2616*x^2 - 2774*x^3 + 1901*x^4 ) / (x-1)^6. - R. J. Mathar, Jul 06 2011

A249163 Triangle read by rows: the positive terms of A163626.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800

Offset: 0

Paul Curtz, Dec 15 2014

We have two possibilities: with or without 0's.
Without 0's:
1,
1,
1,   2,
1,  12,
1,  50,  24,
1, 180, 360,
etc.
Sum of every row: A000670(n).
First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
With 0's:
1,   0,    0,   0,
1,   0,    0,   0,
1,   2,    0,   0,
1,  12,    0,   0,
1,  50,   24,   0,
1, 180,  360,   0,
1, 602, 3360, 720,
etc.
The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)

A368048 a(n) = lcm_{p in Partitions(n)} (Product_{t in p}(t + m)), where m = 2.

Original entry on oeis.org

1, 3, 36, 540, 6480, 136080, 8164800, 24494400, 293932800, 48498912000, 4073908608000, 158882435712000, 9532946142720000, 28598838428160000, 343186061137920000, 612587119131187200000, 7351045429574246400000, 419009589485732044800000, 276546329060583149568000000

Offset: 0

Peter Luschny, Dec 12 2023

nonn

With m = 0, the cumulative radical A048803 is computed, and with m = 1 the Hirzebruch numbers A091137. The general array is A368116. Using the terminology introduced in A368116 a(n) = lcm_{p in P_{2}(n)} Prod(p).

			Let n = 4. The partitions of 4 are [(4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1)]. Thus a(4) = lcm([6, 5*3, 4*4, 4*3*3, 3*3*3*3]) = 6480.

Cf. A368092, A048803, A091137, A368116.

def a(n): return lcm(product(r + 2 for r in p) for p in Partitions(n))
print([a(n) for n in range(20)])

a(n) = A368092(n) * 2^(n - n mod 2).

A368116 A(m, n) = lcm_{p in Partitions(n)} (Product_{r in p}(r + m)). Array read by ascending antidiagonals, for m, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 12, 6, 1, 4, 36, 24, 12, 1, 5, 80, 540, 720, 60, 1, 6, 150, 960, 6480, 1440, 360, 1, 7, 252, 5250, 134400, 136080, 60480, 2520, 1, 8, 392, 1512, 315000, 537600, 8164800, 120960, 5040, 1, 9, 576, 24696, 63504, 1575000, 32256000, 24494400, 3628800, 15120

Offset: 0

Peter Luschny, Dec 12 2023

We say q is a 'm-shifted partition of n' if there is a partition p of n, p = (t1, t2, ..., tk) and q = (t1 + m, t2 + m, ..., tk + m), where m is a nonnegative integer. q is a partition of n + k*m.
Let P(n) denote the partitions of n and P_{m}(n) the m-shifted partitions of n. The product of a partition is the product of its parts, Prod(p) = p1*p2*...*pk if p = (p1, p2, ..., pk). Using this terminology, the definition can be written as A(m, n) = lcm_{p in P_{m}(n)} Prod(p).
With m = 0 the cumulative radical A048803 is computed, and with m = 1 the Hirzebruch numbers A091137.

			Array A(m, n) begins:
  [0] 1, 1,   2,     6,       12,        60,           360, ...  A048803
  [1] 1, 2,  12,    24,      720,      1440,         60480, ...  A091137
  [2] 1, 3,  36,   540,     6480,    136080,       8164800, ...  A368048
  [3] 1, 4,  80,   960,   134400,    537600,      32256000, ...
  [4] 1, 5, 150,  5250,   315000,   1575000,     330750000, ...
  [5] 1, 6, 252,  1512,    63504,   1905120,     880165440, ...
  [6] 1, 7, 392, 24696,  6914880, 532445760,  268352663040, ...
  [7] 1, 8, 576, 23040, 18247680, 145981440,  683193139200, ...
  [8] 1, 9, 810, 80190,  7217100, 844400700, 5851696851000, ...
.
Let m = 2 and n = 4. The partitions of 4 are [(4), (3,1), (2,2), (2,1,1), (1, 1, 1, 1)]. Thus A(2, 4) = lcm([6, 5*3, 4*4, 4*3*3, 3*3*3*3]) = 6480.

Cf. A048803 (m=0), A091137 (m=1), A368048 (m=2).

Columns include: A000027, A011379.

def A(m, n): return lcm(product(r + m for r in p) for p in Partitions(n))
for m in range(9): print([A(m, n) for n in range(7)])

A165886 a(n) = A165641(n+1)/A165641(n).

Original entry on oeis.org

1, 6, 1, 60, 1, 42, 1, 120, 1, 66, 1, 5460, 1, 6, 1, 4080, 1, 798, 1, 660, 1, 138, 1, 10920, 1, 6, 1, 1740, 1, 14322, 1, 8160, 1, 6, 1, 3838380, 1, 6, 1, 54120, 1, 1806, 1, 1380, 1, 282, 1, 371280, 1, 66, 1, 3180, 1, 798, 1, 3480, 1, 354, 1, 113573460, 1, 6, 1, 16320, 1, 64722, 1, 60, 1, 4686, 1, 560403480, 1, 6

Offset: 0

Paul Curtz, Sep 29 2009

nonn

Conjecture: a(n)/A141459(n+1) = A006519(n+1).

The conjecture is correct at least up to n<=2000. - R. J. Mathar, Jul 04 2011

Antti Karttunen, Table of n, a(n) for n = 0..16384

A001316 := proc(n) 2^A000120(n) ;end proc:
A165641 := proc(n) A091137(n)/A001316(n) ;end proc:
A165886 := proc(n) A165641(n+1)/A165641(n) ;end proc:
seq(A165886(n),n=0..60) ; # R. J. Mathar, Jul 04 2011

A001316(n) = 2^hammingweight(n);
A091137(n) = { my(r=1); forprime(p=2, n+1, r*=p^(n\(p-1))); (r); }; \\ From A091137
A165641(n) = (A091137(n)/A001316(n));
A165886(n) = (A165641(n+1)/A165641(n)); \\ Antti Karttunen, Dec 19 2018, after Maple-program

More terms from Antti Karttunen, Dec 19 2018

A363596 a(n) = (Product_{k=1..pi(n+1)} prime(k)^floor(n/(prime(k)-1) ) )/(n+1)!.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 12, 3, 10, 2, 12, 2, 420, 60, 24, 3, 90, 10, 420, 42, 660, 60, 360, 30, 3276, 252, 56, 4, 120, 8, 3696, 231, 3570, 210, 36, 2, 103740, 5460, 840, 42, 13860, 660, 27720, 1260, 19320, 840, 5040, 210, 198900, 7956, 10296, 396, 11880, 440, 6384, 228

Offset: 0

Michael De Vlieger, Aug 03 2023

Motivated by Proposition 3.2, p. 10 of the Bedhouche-Farhi paper.
Observations regarding prime power decomposition of terms in a(0..20737):
For n > 300, most terms are in A361098 but not in A286708. A303606 is a subset of A286708, which is a subset of A361098, which in turn is a subset of A126706, numbers that are neither prime powers nor squarefree.
a(34) = 36 is the only term in A286708 (more specifically, in A303606).
a(35) = 2 is the last prime term.
a(29) = 8 is the only composite prime power.
a(190) = 221760 is the last term in A002182, but a(61) = a(102) = 720720 is the largest.
a(191) = 2310 is the last primorial term.
a(1055) = 2207550414530882190 is the last squarefree term. If there are further squarefree terms a(n), n is likely to belong to -1 (mod 24).
a(7055) = 1733187515208453605856007490304335826298500960 is the last term that is not in A361098. a(n) not in A361098 is likely to belong to -1 (mod 24).

			The table below relates b(n) = A091137(n) to a(n), with (n+1)!*a(n) = k!*m = b(n), where k! is the largest factorial that divides b(n).
 n  A067255(b(n)) (n+1)!*a(n)   k! * m
---------------------------------------
 0  0                1! * 1     1! * 1
 1  1                2! * 1     2! * 1
 2  2.1              3! * 2     3! * 2
 3  3.1              4! * 1     4! * 1
 4  4.2.1            5! * 6     6! * 1
 5  5.2.1            6! * 2     6! * 2
 6  6.3.1.1          7! * 12    7! * 12
 7  7.3.1.1          8! * 3     8! * 3
 8  8.4.2.1          9! * 10   10! * 1
 9  9.4.2.1         10! * 2    10! * 2
10  10.5.2.1.1      11! * 12   12! * 1
11  11.5.2.1.1      12! * 2    12! * 2
12  12.6.3.2.1.1    13! * 420  15! * 2
13  13.6.3.2.1.1    14! * 60   15! * 4
14  14.7.3.2.1.1    15! * 24   15! * 24
15  15.7.3.2.1.1    16! * 3    16! * 3
16  16.8.4.2.1.1.1  17! * 90   18! * 5
...

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See p. 10.
Michael De Vlieger, Log log scatterplot of a(n+1), n = 0..10^4.
Michael De Vlieger, Plot p(k)^e(k) | a(n) at (x, y) = (n, k), n = 0..2^11, with a color function representing e(k), where black = 1, red = 2, and the largest exponent in the dataset shown in magenta. The bar at bottom shows the number 1 in black, primes in red, composite prime powers in gold, squarefree terms in green, and terms that are neither squarefree nor prime powers in blue.

Cf. A000142, A000720, A091137, A361098.

Table[j = 1; ( Times @@ Reap[While[Sow[#^Floor[n/(# - 1)]] &[Prime[j]] > 1, j++]][[-1, 1]] )/Factorial[n + 1], {n, 0, 60}]

from math import prod, factorial
from sympy import sieve
def A363596(n: int) -> int:
    numer = prod(p ** (n // (p - 1)) for p in sieve.primerange(2, n + 2))
    return numer // factorial(n + 1)
print([A363596(n) for n in range(56)])  # Peter Luschny, Aug 17 2025

a(n) = A091137(n)/(n+1)!.

cp's OEIS Frontend

A232193 Numerators of the expected value of the length of a random cycle in a random n-permutation.

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A232248 Denominators of the expected length of a random cycle in a random permutation.

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A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy).

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A145177 Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy).

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A165281 a(n) = (n+1)*(6*n^4 - 51*n^3 + 161*n^2 - 251*n + 251).

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A249163 Triangle read by rows: the positive terms of A163626.

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A368048 a(n) = lcm_{p in Partitions(n)} (Product_{t in p}(t + m)), where m = 2.

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SageMath

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A368116 A(m, n) = lcm_{p in Partitions(n)} (Product_{r in p}(r + m)). Array read by ascending antidiagonals, for m, n >= 0.

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A165281 a(n) = (n+1)(6n^4 - 51n^3 + 161n^2 - 251*n + 251).