A226313 Number of commuting 4-tuples of elements from S_n, divided by n!.
1, 8, 21, 84, 206, 717, 1810, 5462, 13859, 38497, 96113, 253206, 620480, 1566292, 3770933, 9212041, 21768608, 51795427, 120279052, 279849177, 639379257, 1459282932, 3283758256, 7369471795, 16351101855, 36147590987, 79162129897, 172646751524, 373527250619, 804631686843, 1721283389932, 3666041417241
Offset: 1
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..9000
- Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830, 2013
Programs
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Maple
with(numtheory): b:= proc(n) option remember; add(d*sigma(d), d=divisors(n)) end: a:= proc(n) option remember; `if`(n=0, 1, add(add(d* b(d), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=1..40); # Alois P. Heinz, Mar 06 2015 -
Mathematica
b[n_] := b[n] = DivisorSum[n, #*DivisorSigma[1, #]&]; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[j, #*b[#]&]*a[n-j], {j, 1, n}] /n]; Array[a, 40] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *) nmax = 40; Rest[CoefficientList[Series[Exp[Sum[Sum[Sum[d*DivisorSigma[1, d], {d, Divisors[k]}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Mar 31 2018 *) -
PARI
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} seq(n)={my(v=vector(n,i,1)); for(k=1, 2, v=dirmul(v, vector(n,i,i^k))); EulerT(v)} \\ Andrew Howroyd, May 09 2023
Formula
a(n) ~ exp(2^(7/4) * Pi^(3/2) * Zeta(3)^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)) - sqrt(5*Zeta(3)*n) / (2^(3/2)*Pi) + (sqrt(Pi) * 5^(1/4) / (2^(15/4) * 3^(3/2) * Zeta(3)^(1/4)) - sqrt(3) * 5^(5/4) * Zeta(3)^(3/4) / (2^(15/4) * Pi^(7/2))) * n^(1/4) - 25*Zeta(3) / (16*Pi^6) + (5 - 2*Zeta(3)) / (192*Pi^2)) * Pi^(1/4) * Zeta(3)^(1/8) / (2^(13/8) * 3^(1/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 26 2018
Comments