A356004
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d! * (k/d)!).
Original entry on oeis.org
1, 4, 14, 64, 322, 2054, 14380, 116722, 1060580, 10636042, 116996464, 1411275650, 18346583452, 256869465610, 3856674412952, 61743633813634, 1049641774831780, 18896533652098442, 359034139389870400, 7182372973523436802, 150833211474559084844
Offset: 1
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a[n_] := n! * Sum[DivisorSum[k, 1/(#!*(k/#)!) &], {k, 1, n}]; Array[a, 21] (* Amiram Eldar, Jul 22 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k,d,1/(d!*(k/d)!)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(x^k)-1)/k!)/(1-x)))
A381107
Expansion of e.g.f. -log(1-x) * (exp(x) - 1) / (1-x).
Original entry on oeis.org
0, 0, 2, 12, 66, 395, 2665, 20307, 173488, 1646745, 17216653, 196730567, 2440331300, 32666847941, 469457190501, 7210003071247, 117862325748960, 2043420738374545, 37453428525580725, 723643767046525111, 14700326905250293556, 313236372986056228013, 6985951253209713959645
Offset: 0
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a(n) = sum(k=0, n-1, binomial(n, k)*abs(stirling(k+1, 2, 1)));
A381108
Expansion of e.g.f. log(1-x)^2 * (exp(x) - 1) / (2 * (1-x)).
Original entry on oeis.org
0, 0, 0, 3, 30, 245, 2010, 17549, 165942, 1705584, 19024275, 229478689, 2981315139, 41545542818, 618579336284, 9804891730633, 164897938095108, 2933486106772376, 55047126101826453, 1086816606230786217, 22523274090016854661, 488907589907823010158, 11093875133012393113766
Offset: 0
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a(n) = sum(k=0, n-1, binomial(n, k)*abs(stirling(k+1, 3, 1)));
A345887
Number of tilings of an n-cell circular array with rectangular tiles of any size, and where the number of possible colors of a tile is given by the largest cell covered.
Original entry on oeis.org
1, 6, 30, 164, 1030, 7422, 60620, 554248, 5611770, 62353010, 754471432, 9876716940, 139097096918, 2097156230470, 33704296561140, 575219994643472, 10389911153247730, 198019483156015578, 3971390745517868000, 83608226221428800020, 1843561388182505040462
Offset: 1
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a:= proc(n) a(n):= `if`(n=1, 1, a(n-1)*n^2/(n-1)+n) end:
seq(a(n), n=1..21); # Alois P. Heinz, Jun 28 2021
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With[{r = Range[21]}, r*Rest@ FoldList[Times @@ {##} + 1 &, 0, r]] (* Michael De Vlieger, Jun 28 2021 *)
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a(n) = n*sum(k=1, n, n!/k!); \\ Michel Marcus, Jun 29 2021