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

1,2

Comments

Euler transform of A001001.

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

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

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 *)

Prog

(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

Crossrefs

Column k=4 of A362826.

Cf. A001001, A061256, A301777.

Sequence in context: A156304 A232049 A225658 * A275185 A264238 A188700

Adjacent sequences: A226310 A226311 A226312 * A226314 A226315 A226316

Keyword

nonn

Author

N. J. A. Sloane, Jun 08 2013

Status

approved