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%I #40 Feb 13 2024 10:54:05
%S 0,1,5,34,313,3711,54121,937924,18831569,429794605,10987654321,
%T 310989720966,9652968253897,326011399456939,11901025061692313,
%U 466937872906120456,19594541482740368161,875711370981239308953,41524755927216069067489,2082225625247428808306410
%N a(n) = Sum_{m=0..n} m*n^(m-1).
%C The closed form comes from taking the derivative of the closed form of A031972, for which each term of this sequence is a derivative. - _Jonas Whidden_, Oct 18 2011
%H Reinhard Zumkeller, <a href="/A068475/b068475.txt">Table of n, a(n) for n = 0..250</a>
%F a(1) = 1. For n > 1, a(n) = ((n-1)*(n+1)*n^n - n^(n+1) + 1)/(n-1)^2. - _Jonas Whidden_, Oct 18 2011
%F a(n) = A062806(n) / n for n>=1. - _Reinhard Zumkeller_, Nov 22 2014
%F a(n) = [x^(n-1)] 1/((1 - x)*(1 - n*x)^2). - _Peter Bala_, Feb 12 2024
%e a(2) = Sum_{m = 1..2} m*2^(m-1) = 1 + 2*2 = 5.
%p a := n->sum(m*n^(m-1),m=1..n);
%t Join[{0}, Table[Sum[m*n^(m-1), {m,0,n}], {n,1,30}]] (* _G. C. Greubel_, Oct 13 2018 *)
%o (Haskell)
%o a068475 n = sum $ zipWith (*) [1..n] $ iterate (* n) 1
%o -- _Reinhard Zumkeller_, Nov 22 2014
%o (PARI) for(n=0,30, print1(if(n==0, 0, sum(m=0,n, m*n^(m-1))), ", ")) \\ _G. C. Greubel_, Oct 13 2018
%o (Magma) [0] cat [(&+[m*n^(m-1): m in [0..n]]): n in [1..30]]; // _G. C. Greubel_, Oct 13 2018
%Y Derivative sequence of A031972.
%Y Cf. A023037, A062805, A062806, A368534.
%K nonn,easy
%O 0,3
%A Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002
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