A354845
a(n) = n! * Sum_{d|n} (n/d)^(d-1) / d!.
Original entry on oeis.org
1, 3, 7, 49, 121, 2281, 5041, 134401, 907201, 13184641, 39916801, 3753509761, 6227020801, 393409336321, 7638997766401, 160474477363201, 355687428096001, 75792615407308801, 121645100408832001, 32459310892353945601, 475723576423839744001, 7306033564948620902401
Offset: 1
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a[n_] := n! * DivisorSum[n, (n/#)^(#-1)/#! &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)
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a(n) = n!*sumdiv(n, d, (n/d)^(d-1)/d!);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(k*x^k)-1)/k)))
A327579
a(n) = n! * Sum_{d|n} d^(n/d) / d!.
Original entry on oeis.org
1, 4, 9, 76, 125, 4686, 5047, 389768, 1995849, 62445610, 39916811, 23574862092, 6227020813, 5667436494734, 55630647072015, 2922249531801616, 355687428096017, 2425220588831040018, 121645100408832019, 1364553980880330240020, 18677216386213152768021, 1152100749379237026969622
Offset: 1
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a[n_] := n! Sum[d^(n/d)/d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/((k - 1)! (1 - k x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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a(n) = n! * sumdiv(n, d, d^(n/d) / d!); \\ Michel Marcus, Sep 17 2019
A356668
Expansion of e.g.f. Sum_{k>=0} x^k / (k! - k*x^k).
Original entry on oeis.org
1, 1, 3, 7, 37, 121, 1141, 5041, 60761, 378001, 5444461, 39916801, 729041545, 6227020801, 130767460825, 1321314894901, 31388220966961, 355687428096001, 9636906872926477, 121645100408832001, 3649432697160095561, 51223991519836175041, 1686001091666419279753
Offset: 0
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a[n_]:= n! * DivisorSum[n, 1/(# * (# - 1)!^(n/#)) &]; a[0] = 1; Array[a, 23, 0] (* Amiram Eldar, Aug 22 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!-k*x^k))))
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a(n) = if(n==0, 1, n!*sumdiv(n, d, 1/(d*(d-1)!^(n/d))));
A356674
a(n) = n! * Sum_{k=0..n} k^(k*(n-k))/k!.
Original entry on oeis.org
1, 2, 5, 25, 349, 19941, 4440391, 4382699203, 17687865017481, 356274213630958297, 33338407933090938442411, 16214021627369697901867402911, 43817834057167927861655409052462093, 595284492835035398061242850538179192931525
Offset: 0
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Table[n!*(1 + Sum[k^(k*(n-k))/k!, {k, 1, n}]), {n, 0, 12}] (* Vaclav Kotesovec, Nov 27 2022 *)
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a(n) = n!*sum(k=0, n, k^(k*(n-k))/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k^k*x)))))
A354889
a(n) = n! * Sum_{d|n} d^(d-1) / d!.
Original entry on oeis.org
1, 4, 15, 112, 745, 10296, 122689, 2285312, 43953921, 1026157600, 25977341401, 751135431168, 23304312143281, 795924137531264, 29203006015310625, 1154107395053387776, 48661547563094964481, 2186762596692631699968, 104127471943011650364841
Offset: 1
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a[n_] := n! * DivisorSum[n, #^(# - 1)/#! &]; Array[a, 19] (* Amiram Eldar, Jun 10 2022 *)
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a(n) = n!*sumdiv(n, d, d^(d-1)/d!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k^(k-1)*x^k/(k!*(1-x^k)))))
A330020
Expansion of e.g.f. Sum_{k>=1} x^k / (k! * (1 - x^k)^k).
Original entry on oeis.org
1, 3, 7, 49, 121, 2161, 5041, 127681, 725761, 12852001, 39916801, 2917918081, 6227020801, 392423391361, 4740319584001, 122053759027201, 355687428096001, 57808258040332801, 121645100408832001, 18854997267794688001, 289799177540640768001, 7306005040298918553601
Offset: 1
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nmax = 22; CoefficientList[Series[Sum[x^k/(k! (1 - x^k)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := n! Sum[(d + n/d - 2)!/(d! (d - 1)! (n/d - 1)!), {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
A357296
Expansion of e.g.f. Sum_{k>0} x^k / (k! * (1 - x^k/k)).
Original entry on oeis.org
1, 3, 7, 31, 121, 851, 5041, 43261, 369601, 3748249, 39916801, 490801081, 6227020801, 87861842641, 1310800947457, 21018206008801, 355687428096001, 6419518510204801, 121645100408832001, 2435836129700029057, 51102829650622464001, 1124549558817839481601
Offset: 1
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a[n_] := n! * DivisorSum[n, 1/(#^(n/#-1) * #!) &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k/k)))))
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a(n) = n!*sumdiv(n, d, 1/(d^(n/d-1)*d!));
Showing 1-7 of 7 results.