A363697
a(n) = -n! * Sum_{d|n} (-n/d)^d / d!.
Original entry on oeis.org
1, 3, 19, 47, 601, 2039, 35281, -26881, 4898881, -8104321, 439084801, 576132479, 80951270401, -913158005761, 49506372115201, -558073906790401, 6046686277632001, 79958674981785599, 2311256907767808001, -115583806104986419201
Offset: 1
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a[n_] := -n! * DivisorSum[n, (-n/#)^#/#! &]; Array[a, 20] (* Amiram Eldar, Jul 03 2023 *)
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a(n) = -n!*sumdiv(n, d, (-n/d)^d/d!);
A363698
a(n) = n! * Sum_{d|n} (-1)^(d+1) * (n/d)^n / d!.
Original entry on oeis.org
1, 7, 163, 5951, 375001, 33337559, 4150656721, 675135713279, 140588337476161, 36270281280965759, 11388728893445164801, 4270306368140557557119, 1886009588552176549862401, 968696203690612910273080319
Offset: 1
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a[n_] := n! * DivisorSum[n, (-1)^(#+1) * (n/#)^n / #! &]; Array[a, 15] (* Amiram Eldar, Jul 03 2023 *)
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a(n) = n!*sumdiv(n, d, (-1)^(d+1)*(n/d)^n/d!);
A336242
a(n) = (n!)^2 * Sum_{d|n} (-1)^(d+1) / (d!)^2.
Original entry on oeis.org
1, 3, 37, 431, 14401, 403199, 25401601, 1216454399, 135339724801, 9877056537599, 1593350922240001, 178056522962841599, 38775788043632640001, 5700041141609893478399, 1757631343928533032960001, 327562346808114783805439999, 126513546505547170185216000001
Offset: 1
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Table[(n!)^2 Sum[(-1)^(d + 1)/(d!)^2, {d, Divisors[n]}], {n, 1, 17}]
nmax = 17; CoefficientList[Series[Sum[(1 - BesselJ[0, 2 x^(k/2)]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2 // Rest
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a(n) = n!^2*sumdiv(n, d, (-1)^(d+1)/d!^2); \\ Michel Marcus, Jul 13 2020
A354022
a(n) = n! * Sum_{d|n} mu(n/d) / d!.
Original entry on oeis.org
1, -1, -5, -11, -119, 241, -5039, -1679, -60479, 1784161, -39916799, 218877121, -6227020799, 43571848321, 1078831353601, -518918399, -355687428095999, 1058152455360001, -121645100408831999, 1115079416638387201, 42565648051390464001, 562000335730215782401
Offset: 1
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Table[n! Sum[MoebiusMu[n/d]/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] (Exp[x^k] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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a(n)=n! * sumdiv(n, d, moebius(n/d) / d!) \\ Winston de Greef, Sep 19 2023