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!)))
A351182
a(n) = Sum_{k=0..n} k^(2*k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 683, 61332, 9135004, 2035708760, 634172615600, 263166948202080, 140322186951905736, 93484350581344936344, 76095870609142447018152, 74311960997497053384537408, 85748280952260853814490688656
Offset: 0
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a(n) = sum(k=0, n, k^(2*k)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*log(1+x))^k/k!)))
A373858
a(n) = Sum_{k=1..n} k! * k^(2*n-1) * Stirling1(n,k).
Original entry on oeis.org
0, 1, 15, 1268, 317294, 175542694, 181641609214, 315309390376056, 850661260866748728, 3370191684116333977872, 18768704088141613880906736, 141902519646656406912522712848, 1415862822521619228707500717132224, 18210234893009450819658863637633454608
Offset: 0
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nmax=13; Range[0,nmax]!CoefficientList[Series[Sum[(Log[1 + k^2*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^(2*n-1)*stirling(n, k, 1));
A373861
a(n) = Sum_{k=0..n} k^(2*n) * |Stirling1(n,k)|.
Original entry on oeis.org
1, 1, 17, 923, 107724, 22369324, 7385651720, 3597082257152, 2449105468081600, 2238708422118782376, 2661994302285967390224, 4014423110086784061347592, 7519716937006429200213786240, 17194081369411703462470895338272
Offset: 0
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Unprotect[Power]; Power[0, 0] = 1; Protect[Power]; nmax=13; Range[0,nmax]!CoefficientList[Series[Sum[(-Log[1 - k^2*x])^k / k!,{k,0,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 19 2024 *)
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a(n) = sum(k=0, n, k^(2*n)*abs(stirling(n, k, 1)));
Showing 1-4 of 4 results.