A351180
a(n) = Sum_{k=0..n} k^(k+n) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 635, 53112, 7367444, 1529130770, 443685287576, 171495189203456, 85174828026304824, 52856314387144232184, 40077340463437963801752, 36457068309928364981668848, 39186634107857517367884040632
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)
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a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*log(1+k*x))^k/k!)))
A351183
a(n) = Sum_{k=0..n} k^(2*n) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 539, 28980, 1295404, -177715720, -88870557952, -11213754156480, 11072302541223336, 8352732988619491824, -1800044600955923261688, -8483589341410812834791040, -2945489916041839476122254560
Offset: 0
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a(n) = sum(k=0, n, k^(2*n)*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^2*x)^k/k!)))
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));
A350726
a(n) = Sum_{k=0..n} k^(n-k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 0, -3, 21, -100, -525, 33026, -860503, 16304464, -100885935, -12798492630, 1037135603845, -55556702499792, 2207903148318777, -31916679640973750, -6164889702150516015, 983802138243128355456, -100629406324320358067423
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(n - k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 19, 0] (* Amiram Eldar, Feb 03 2022 *)
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a(n) = sum(k=0, n, 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^k))))
A383051
a(n) is the n-th term of the inverse Stirling transform of j-> (j+1)^n.
Original entry on oeis.org
1, 2, 5, -1, -116, 984, 16400, -788418, 5474016, 941115360, -51647682648, -264087895512, 244846563852864, -16953959408998080, -436871956049596800, 219647419965976413744, -20283048895473275917824, -877465277974899660349440, 545297904370739513319183360
Offset: 0
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a(n) = sum(k=0, n, (k+1)^n*stirling(n, k, 1));
Showing 1-5 of 5 results.