A076726 a(n) = Sum_{k>=0} k^n/2^k.

2, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646

Offset: 0

Author

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

Keywords

• nonn

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
• Ramesh L. Srigiriraju, Recurrences for A076726

Crossrefs

Same as A000629 except for a(0).

A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

Programs

Mathematica

a[n_] := Sum[(k^n)/(2^k), {k, 0, Infinity}]; Table[ a[n], {n, 0, 18}]
a[n_] := (-1)^(n+1) PolyLog[-n, 2] (* Vladimir Reshetnikov, Jan 23 2011 *)

PARI

a(n)=abs(polylog(-n, 2)) \\ Charles R Greathouse IV, Jul 15 2014

Formula

a(n) = 2*A000670(n). - Philippe Deléham, Mar 06 2004

a(n) ~ n! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 28 2013

From Jianing Song, May 04 2022: (Start)

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

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

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

Extensions

More terms from Robert G. Wilson v, Oct 29 2002