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A226313
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Number of commuting 4-tuples of elements from S_n, divided by n!.
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4
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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
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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];
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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