A014143 Partial sums of A014138.

1, 4, 12, 34, 98, 294, 919, 2974, 9891, 33604, 116103, 406614, 1440025, 5147876, 18550572, 67310938, 245716094, 901759950, 3325066996, 12312494462, 45766188948, 170702447074, 638698318850, 2396598337950

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Self-convolution of A014137. Column in triangle A200965. - Philippe Deléham, Jan 24 2014

For n >= 2, a(n-2) is the number of 021-avoiding ascent sequences of length n with exactly one occurrence of the consecutive pattern 01. For example, with n=3, a(1)=4 counts 001, 010, 011, 012. - David Callan, Nov 13 2019

References

• Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012. - From _N. J. A. Sloane_, May 09 2012 [An early version on the arXiv had A014043 instead of A014143]
• Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv, arXiv:1302.2274 [math.CO], 2013)

Crossrefs

Cf. A014137, A200965.

Programs

Mathematica

Table[SeriesCoefficient[(1-2*x-Sqrt[1-4*x])/(2*x^2*(1-x)^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
Table[2*Sum[Sum[Binomial[2*j+1,j]/(j+2),{j,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 27 2012 *)

PARI

x='x+O('x^66); Vec((1-2*x-sqrt(1-4*x))/(2*x^2*(1-x)^2)) \\ Joerg Arndt, May 04 2013

Formula

G.f.: (1-2*z-sqrt(1-4*z))/(2*z^2*(1-z)^2). - Emeric Deutsch, Jan 27 2003

Recurrence: (n+2)*a(n) = 6*(n+1)*a(n-1) - 3*(3*n+2)*a(n-2) + 2*(2*n+1)*a(n-3). - Vaclav Kotesovec, Oct 07 2012

a(n) ~ 2^(2n+6)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012

a(n) = 2 * Sum_{k=0..n} Sum_{j=0..k} C(2*j+1,j)/(j+2). - Vaclav Kotesovec, Oct 27 2012