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a(n) = Sum_{k=0..n} binomial(n-1,k-1)*k^n*n!/k!; a(0) = 1.
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%I #20 Jun 15 2022 07:50:14

%S 1,1,6,81,1828,60565,2734926,160109005,11724156648,1045312448841,

%T 111114793839610,13845807451708441,1994597720747571468,

%U 328351264019737949341,61162428777982281583302,12782305566531823350524805,2975150384583838798131401296,766253903501365584725344992529

%N a(n) = Sum_{k=0..n} binomial(n-1,k-1)*k^n*n!/k!; a(0) = 1.

%C a(n) is the n-th term of the Lah transform of the n-th powers.

%H G. C. Greubel, <a href="/A317277/b317277.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 A317277:= n-> `if`(n=0,1, add(binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n)); seq(A317277(n), n=0..30); # _G. C. Greubel_, Mar 09 2021

%t Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^n n!/k!, {k, n}], {n, 17}]]

%t Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 - x))^k/k!, {k, n}], {x, 0, n}], {n, 17}]]

%o (Sage) [1]+[sum(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[(&+[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, binomial(n-1, k-1)*k^n*n!/k!)); \\ _Michel Marcus_, Mar 10 2021; corrected Jun 15 2022

%Y Cf. A000262, A052852, A103194, A293145, A317278, A317279.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 25 2018

%E Name edited by _Michel Marcus_, Jun 15 2022