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%I #17 Jun 13 2022 03:01:42
%S 1,1,2,-15,-164,4245,46386,-4901939,39141656,11707820361,
%T -671114863610,-29398709945319,7385525824325364,-307076643365636963,
%U -73748845974115224262,14299745046516639280005,-237996466462017367478864,-377740669670216316717155055,75515477307532501838072029326
%N a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*k^n*n!/k!.
%C a(n) is the n-th term of the inverse Lah transform of the n-th powers.
%H G. C. Greubel, <a href="/A317278/b317278.txt">Table of n, a(n) for n = 0..250</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = n! * [x^n] Sum_{k>=0} k^n*(x/(1 + x))^k/k!.
%p A317278:= n-> `if`(n=0,1, add((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n));
%p seq(A317278(n), n=0..30); # _G. C. Greubel_, Mar 09 2021
%t Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] k^n n!/k!, {k, n}], {n, 18}]]
%t Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 + x))^k/k!, {k, n}], {x, 0, n}], {n, 18}]]
%o (Sage) [1]+[sum((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # _G. C. Greubel_, Mar 09 2021
%o (Magma) [1]cat[(&+[(-1)^(n+j)*Binomial(n-1,j-1)*Binomial(n,j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // _G. C. Greubel_, Mar 09 2021
%o (PARI) a(n) = if (n==0, 1, sum(k=0, n, (-1)^(n-k)*binomial(n-1, k-1)*k^n*n!/k!)); \\ _Michel Marcus_, Mar 10 2021; corrected Jun 13 2022
%Y Cf. A111884, A256467, A317277, A317279.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jul 25 2018
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