A373857
a(n) = Sum_{k=1..n} k! * k^(n-1) * Stirling1(n,k).
Original entry on oeis.org
0, 1, 3, 32, 734, 28994, 1752046, 150262104, 17356844088, 2597710341600, 488957612319984, 113044488306692304, 31490845086661001664, 10403092187976909854640, 4021236906890850070201488, 1798052050351216209712206336, 920859156623446912386646303104
Offset: 0
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nmax=16; Range[0,nmax]!CoefficientList[Series[Sum[(Log[1 + k*x])^k / k,{k,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 19 2024 *)
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a(n) = sum(k=1, n, k!*k^(n-1)*stirling(n, k, 1));
A373874
a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling1(n,k).
Original entry on oeis.org
0, 1, 1, 8, 142, 4534, 229658, 16951416, 1718394312, 229119947280, 38881745126112, 8183542269446928, 2092128552508587360, 638590833851037194256, 229398149222697428624688, 95801846241560025353728512, 46025711723325944648182502016
Offset: 0
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a(n) = sum(k=1, n, k!*k^(n-2)*stirling(n, k, 1));
A350725
a(n) = Sum_{k=0..n} k! * k^(n-k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 1, -4, -2, 274, -3442, -12552, 2108664, -63083232, 87416112, 112192496976, -7487840132544, 174521224997040, 19793498724358032, -3109195219736188416, 209306170972547346816, 2973238556525799866496, -3013574861684426837113728, 456220653756733889826621696
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * k^(n-k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2022 *)
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a(n) = sum(k=0, n, k!*k^(n-k)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k*x)^k/k^k)))