A139756 Binomial transform of A004526.

0, 0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, 11264, 24576, 53248, 114688, 245760, 524288, 1114112, 2359296, 4980736, 10485760, 22020096, 46137344, 96468992, 201326592, 419430400, 872415232, 1811939328, 3758096384, 7784628224, 16106127360, 33285996544

Offset: 0

Author

Paul Curtz, May 19 2008

Keywords

• nonn
• easy

Comments

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

Also: self-convolution of A131577. - R. J. Mathar, May 22 2008

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.

Examples

			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.
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

a(n) = A097067(n), n>0. - R. J. Mathar, Nov 03 2008

a(n) = A168511(n+1,n). - Philippe Deléham, Mar 20 2013

a(n) = 2*a(n-1) + 2^(n-2), n>=2. - Philippe Deléham, Mar 20 2013

Extensions

More terms from R. J. Mathar, May 22 2008