%I #49 Jun 04 2024 16:53:43

%S 2,2,6,26,150,1082,9366,94586,1091670,14174522,204495126,3245265146,

%T 56183135190,1053716696762,21282685940886,460566381955706,

%U 10631309363962710,260741534058271802,6771069326513690646

%N a(n) = Sum_{k>=0} k^n/2^k.

%H Vincenzo Librandi, <a href="/A076726/b076726.txt">Table of n, a(n) for n = 0..200</a>

%H Olivier Golinelli, <a href="https://arxiv.org/abs/2405.16968">Remote control system of a binary tree of switches - II. balancing for a perfect binary tree</a>, arXiv:2405.16968 [cs.DM], 2024. See p. 17.

%H Ramesh L. Srigiriraju, <a href="/A076726/a076726.pdf">Recurrences for A076726</a>

%F a(n) = 2*A000670(n). - _Philippe Deléham_, Mar 06 2004

%F a(n) ~ n! / (log(2))^(n+1). - _Vaclav Kotesovec_, Nov 28 2013

%F From _Jianing Song_, May 04 2022: (Start)

%F a(0) = 2, a(n) = Sum_{k=0..n-1} binomial(n,k)*a(k) for n >= 1.

%F G.f.: Sum_{k>=0} 1/(2^k*(1-k*x)).

%F E.g.f.: 1/(1-exp(x)/2). (End)

%e a(0) = 2 because 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 2; a(1) = 2 because 0 + 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + ... = 2.

%e G.f. = 2 + 2*x + 6*x^2 + 26*x^3 + 150*x^4 + 1082*x^5 + 9366*x^6 + 94586*x^7 + ...

%t a[n_] := Sum[(k^n)/(2^k), {k, 0, Infinity}]; Table[ a[n], {n, 0, 18}]

%t a[n_] := (-1)^(n+1) PolyLog[-n, 2] (* _Vladimir Reshetnikov_, Jan 23 2011 *)

%o (PARI) a(n)=abs(polylog(-n, 2)) \\ _Charles R Greathouse IV_, Jul 15 2014

%Y Same as A000629 except for a(0).

%Y A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - _N. J. A. Sloane_, Jul 04 2012

%K nonn

%O 0,1

%A Charles G. Waldman (cgw(AT)alum.mit.edu), Oct 27 2002

%E More terms from _Robert G. Wilson v_, Oct 29 2002