A087132 -id:A087132 - cp's OEIS frontend

A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

1, 1, 1, 13, 100, 1876, 57636, 2051316, 104640768, 6819033600, 576652089600, 57187381536000, 7057192160793600, 1014733052692300800, 172646881540527744000, 33848454886497227289600, 7637231669166956976537600, 1948418678155880277481881600

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

			a(n) ~ c * (n-1)!^2, where c = A156648 = Product_{k>=1} (1 + 1/k^2) = sinh(Pi)/Pi = 3.67607791037497772...

Alois P. Heinz, Table of n, a(n) for n = 0..254

Cf. A007838, A249588, A326865.

Cf. A087132.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2023

nmax = 20; CoefficientList[Series[Product[(1+x^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!^2

A206820 a(n) is the sum of the squares of the sizes of the conjugacy classes in the symmetric group A_n.

Original entry on oeis.org

1, 1, 3, 42, 914, 23694, 1048542, 45379878, 3272115926, 257662344206, 27726935045366, 3101635433302996, 474878584235678020, 76786899439922296204, 15844064187141655171020, 3326909755872288926885670, 897661138669999282018222470, 246381314116108359863665821750

Offset: 1

Olivier Gérard, Feb 12 2012

a(n) is the sum over all elements of Alt_n of the size of their conjugacy class. Each conjugacy class is thus counted as many times as its size, giving a sum of squares.

			For n=5, a(5) = 1 + 12^2 + 12^2 + 15^2 + 20^2 = 914.
The class equation of A_5 is  1 + 12 + 12 + 15 + 20  = 60 = 5!/2

A087132 (sequence for S_n), A000702 (conjugacy classes in A_n)

A206820 := n -> Sum(ConjugacyClasses(AlternatingGroup(n)), c->Size(c)^2); # Eric M. Schmidt, Jan 26 2014

More terms from Eric M. Schmidt, Jan 26 2014

A192983 a(n) is the number of pairs (g, h) of elements of the symmetric group S_n such that g and h have conjugates that commute.

Original entry on oeis.org

1, 4, 24, 264, 5640, 151200, 5722920, 282868992, 18371308032, 1504791561600, 148978034686800, 18007146260231040, 2528615024682544512, 426310052282058252672, 81830910530970671616000, 18305445786667543107072000, 4570435510076312321728158720

Offset: 1

Mark Wildon, Aug 03 2011

nonn

a(n) / n!^2 is the probability that two permutation in S_n, chosen independently and uniformly at random, have conjugates that commute.

Apparently n | a(n), and, for n>1, n*(n-1) | a(n). - Alexander R. Povolotsky, Sep 30 2011

			For n = 3 the probability that two elements of S_3 have conjugates that commute is a(3)/3!^2 = 2/3. Proof: only the transpositions and three cycles fail to have conjugates that commute; the probability of choosing one permutation from each of these classes is 2*1/2*1/3 = 1/3.

Simon R. Blackburn, John R. Britnell, and Mark Wildon, The probability that a pair of elements of a finite group are conjugate, arXiv:1108.1784 [math.GR], 2011-2012.
J. R. Britnell and M. Wildon, Commuting elements in conjugacy classes: an application of Hall's Marriage Theorem to group theory, J. Group Theory, 12 (2009), 795-802.
Mark Wildon, Haskell source code for computing values of the sequence.

Cf. A087132 (the sum of squares of the sizes of the conjugacy classes of S_n).

Haskell
```
-- See links for code.
```

A246879 Decimal expansion of the constant W(1) appearing in the asymptotic expression of the probability that two independent, random n-permutations have the same cycle type as W(1)/n^2.

Original entry on oeis.org

4, 2, 6, 3, 4, 0, 3, 5, 1, 4, 1, 5, 2, 6, 6, 9, 7, 7, 8, 2, 9, 8, 9, 3, 5, 0, 5, 5, 1, 6, 6, 1, 9, 6, 6, 9, 0, 5, 3, 5, 0, 8, 1, 8, 1, 7, 4, 7, 9, 4, 1, 1, 6, 0, 5, 0, 6, 7, 7, 1, 2, 5, 6, 3, 2, 0, 3, 7, 1, 9, 1, 4, 5, 8, 2, 7, 8, 5, 7, 3, 4, 6, 1, 7, 2, 3, 5, 6, 1, 3, 4, 4, 8, 1, 3, 2, 9, 8, 7, 7, 3, 0, 6, 3, 5

Offset: 1

Jean-François Alcover, Sep 08 2014

See A087132.

			4.2634035141526697782989350551661966905350818174794...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 42.
Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, N. Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO]

Cf. A087132.

evalf(product(BesselI(0,2/k), k=1..infinity), 100) # Vaclav Kotesovec, Sep 17 2014

digits = 50; m0 = 1000; dm = 1000; tail[m_] := PolyGamma[1, m] - (1/24)*PolyGamma[3, m] + PolyGamma[5, m]/1080 - (11*PolyGamma[7, m])/967680 + (19*PolyGamma[9, m])/217728000 - (43*PolyGamma[11, m])/94058496000; Clear[f]; f[m_] := f[m] = Sum[Log[BesselI[0, 2/k]], {k, 1, m - 1}] + tail[m] // N[#, digits + 5] &; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 2] != RealDigits[f[m - dm], 10, digits + 2], Print["f(", m, ") = ", f[m]]; m = m + dm];RealDigits[Exp[f[m]], 10, digits] // First

prod_{k>=1} I_0(2/k), where I_0 is the zeroth modified Bessel function.

More terms from Vaclav Kotesovec, Sep 17 2014

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A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

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A206820 a(n) is the sum of the squares of the sizes of the conjugacy classes in the symmetric group A_n.

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A192983 a(n) is the number of pairs (g, h) of elements of the symmetric group S_n such that g and h have conjugates that commute.

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Haskell

A246879 Decimal expansion of the constant W(1) appearing in the asymptotic expression of the probability that two independent, random n-permutations have the same cycle type as W(1)/n^2.

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