A139756 - OEIS

%I #29 May 06 2024 12:10:25

%S 0,0,1,4,12,32,80,192,448,1024,2304,5120,11264,24576,53248,114688,

%T 245760,524288,1114112,2359296,4980736,10485760,22020096,46137344,

%U 96468992,201326592,419430400,872415232,1811939328,3758096384,7784628224,16106127360,33285996544

%N Binomial transform of A004526.

%C Essentially the same as A001787, A097067, A085750 and A118442.

%C Also: self-convolution of A131577. - _R. J. Mathar_, May 22 2008

%C Let S be a subset of {1,2,...,n}.  A succession in S is a subset of the form {i,i+1}.  a(n) is the total number of successions in all subsets of {1,2,...,n}.  a(n) = Sum_{k>=1} A076791(n,k)*k. - _Geoffrey Critzer_, Mar 18 2012.

%D I Goulden and D Jackson, Combinatorial Enumeration, John Wiley and Sons, 1983, page 55.

%H Vincenzo Librandi, <a href="/A139756/b139756.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).

%F O.g.f.: x^2/(1-2*x)^2. a(n) = (n-1)*2^n/4 if n>0. - _R. J. Mathar_, May 22 2008

%F a(n) = A097067(n), n>0. - _R. J. Mathar_, Nov 03 2008

%F a(n) = A168511(n+1,n). - _Philippe Deléham_, Mar 20 2013

%F a(n) = 2*a(n-1) + 2^(n-2), n>=2. - _Philippe Deléham_, Mar 20 2013

%e a(4) = 12 because we have {1,2}, {2,3}, {3,4}, {1,2,4}, {1,3,4} with one succession; {1,2,3}, {2,3,4} with two successions; and {1,2,3,4} with three successions. - _Geoffrey Critzer_, Mar 18 2012.

%t nn = 30; a = 1/(1 - y x); b = x/(1 - y x) + 1; c = 1/(1 - x); CoefficientList[ D[Series[c b/(1 - (a x^2 c)), {x, 0, nn}], y] /. y -> 1, x]  (* _Geoffrey Critzer_, Mar 18 2012 *)

%Y Cf. A001787, A004526, A076791, A085750, A097067, A118442, A131577, A168511.

%K nonn,easy

%O 0,4

%A _Paul Curtz_, May 19 2008

%E More terms from _R. J. Mathar_, May 22 2008