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

0,2

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)

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

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 *)

Prog

(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

Crossrefs

Cf. A014137, A200965.

Sequence in context: A079818 A115390 A005056 * A361476 A077994 A077843

Adjacent sequences: A014140 A014141 A014142 * A014144 A014145 A014146

Keyword

nonn

Author

N. J. A. Sloane

Status

approved