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A133286 a(n) is the difference by which n^n overestimates the value of (1/2) Sum_{k>=0} k^n/2^k.

%I A133286 #29 Nov 10 2020 15:44:23
%S A133286 0,0,1,14,181,2584,41973,776250,16231381,380333228,9897752437,
%T A133286 283689038038,8888008880661,302348248243872,11101365482587573,
%U A133286 437663607189881522,18441428419027570261,827109891119307628276,39343022540633280730101,1978326854072994260712846
%N A133286 a(n) is the difference by which n^n overestimates the value of (1/2) Sum_{k>=0} k^n/2^k.
%H A133286 Alois P. Heinz, <a href="/A133286/b133286.txt">Table of n, a(n) for n = 0..380</a>
%F A133286 a(n) = n^n - (1/2) Sum_{k>=0} k^n/2^k.
%F A133286 a(n) = A000312(n) - A000670(n). - _Alois P. Heinz_, Jul 29 2014
%F A133286 E.g.f.: 1/(1+LambertW(-x)) - 1/(2-exp(x)). - _Alois P. Heinz_, Aug 03 2014
%e A133286 a(3) = 3^3 - (1/2) Sum_{k>=0} k^3/2^k = 27 - 1/2 * 26 = 27 - 13 = 14.
%p A133286 a:= n-> n^n -sum(k^n/2^k, k=0..infinity)/2:
%p A133286 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 29 2014
%p A133286 # second Maple program:
%p A133286 b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
%p A133286 a:= n-> n^n- b(n)*n!:
%p A133286 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 29 2014
%t A133286 a[n_] := If[n==0, 0, n^n - HurwitzLerchPhi[1/2, -n, 0]/2];
%t A133286 a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 10 2020 *)
%Y A133286 Cf. A000312, A000670.
%Y A133286 Cf. column k=0 of A245733.
%K A133286 nonn
%O A133286 0,4
%A A133286 Ramesh L. Srigiriraju (rsrigir(AT)vt.edu), Oct 16 2007
%E A133286 More terms and a(14)-a(17) corrected by _Alois P. Heinz_, Jul 29 2014