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

Offset

0,4

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=1,2,...} A076791(n,k)*k. - 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).

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. [From 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

Example

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.

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*)

Crossrefs

Sequence in context: A097392 A090634 A260186 * A085750 A001787 A118442

Adjacent sequences: A139753 A139754 A139755 * A139757 A139758 A139759

Keyword

nonn,easy

Author

Paul Curtz, May 19 2008

Extensions

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

Status

approved