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

0, 0, 1, 14, 181, 2584, 41973, 776250, 16231381, 380333228, 9897752437, 283689038038, 8888008880661, 302348248243872, 11101365482587573, 437663607189881522, 18441428419027570261, 827109891119307628276, 39343022540633280730101, 1978326854072994260712846

Offset: 0

Author

Ramesh L. Srigiriraju (rsrigir(AT)vt.edu), Oct 16 2007

Keywords

• nonn

Examples

			a(3) = 3^3 - (1/2) Sum_{k>=0} k^3/2^k = 27 - 1/2 * 26 = 27 - 13 = 14.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A000312, A000670.

Cf. column k=0 of A245733.

Programs

Maple

a:= n-> n^n -sum(k^n/2^k, k=0..infinity)/2:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2014
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
a:= n-> n^n- b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2014

Mathematica

a[n_] := If[n==0, 0, n^n - HurwitzLerchPhi[1/2, -n, 0]/2];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 10 2020 *)

Formula

a(n) = n^n - (1/2) Sum_{k>=0} k^n/2^k.

a(n) = A000312(n) - A000670(n). - Alois P. Heinz, Jul 29 2014

E.g.f.: 1/(1+LambertW(-x)) - 1/(2-exp(x)). - Alois P. Heinz, Aug 03 2014

Extensions

More terms and a(14)-a(17) corrected by Alois P. Heinz, Jul 29 2014