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%I #24 Sep 09 2024 16:40:19
%S 0,5,50,375,2500,15625,93750,546875,3125000,17578125,97656250,
%T 537109375,2929687500,15869140625,85449218750,457763671875,
%U 2441406250000,12969970703125,68664550781250,362396240234375,1907348632812500,10013580322265625,52452087402343750
%N a(n) = n*5^n.
%H Vincenzo Librandi, <a href="/A036291/b036291.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-25).
%F G.f.: 5*x/(1 - 5*x)^2. - _Vincenzo Librandi_, Sep 09 2014
%F From _Amiram Eldar_, Jul 20 2020: (Start)
%F Sum_{n>=1} 1/a(n) = log(5/4).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(6/5). (End)
%F From _Elmo R. Oliveira_, Sep 09 2024: (Start)
%F E.g.f.: 5*x*exp(5*x).
%F a(n) = n*A000351(n) = 5*A053464(n).
%F a(n) = 10*a(n-1) - 25*a(n-2) for n > 1. (End)
%p g:=1/(1-5*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)*n, n=0..31); # _Zerinvary Lajos_, Jan 09 2009
%t Table[n 5^n, {n, 0, 20}] (* _Vincenzo Librandi_, Sep 09 2014 *)
%o (Magma) [n*5^n: n in [0..20]]; // _Vincenzo Librandi_, Sep 09 2014
%Y Cf. A000351, A053464.
%K nonn,easy,changed
%O 0,2
%A _N. J. A. Sloane_
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