A212346 Sequence of coefficients of x^0 in marked mesh pattern generating function Q_{n,132}^(0,4,0,0)(x).

1, 1, 2, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733

Offset: 0

Author

N. J. A. Sloane, May 09 2012

Keywords

• nonn
• more

Comments

Conjecture stated in Formula holds through a(35).

Links

• S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243, 2012

Programs

Mathematica

QQ0[t, x] = (1 - (1-4*x*t)^(1/2)) / (2*x*t); QQ1[t, x] = 1/(1 - t*QQ0[t, x]); QQ2[t, x] = (1 + t*(QQ1[t, x] - QQ0[t, x]))/(1 - t*QQ0[t, x]); QQ3[t, x] = (1 + t*(QQ2[t, x] - QQ0[t, x] + t*(QQ1[t, x] - QQ0[t,  x])))/(1 - t*QQ0[t, x]); QQ4[t, x] = (1 + t*(QQ3[t, x] - QQ0[t, x] + t*(QQ2[t, x] - QQ0[t, x]) + (2*t^2*(QQ1[t, x] - QQ0[t, x]))))/(1 - t*QQ0[t, x]); q=Simplify[Series[QQ4[t, x], {t, 0, 35}]]; CoefficientList[q /. x -> 0, t]  (* Robert Price, Jun 04 2012 *)

Formula

Conjecture: this appears to equal (n+3)(n^2-4)/6 for n >= 3, see A129936.

Conjectures from Colin Barker, Feb 11 2015: (Start)

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>6.

G.f.: (2*x^6-5*x^5+3*x^4-x^3+4*x^2-3*x+1) / (x-1)^4.

(End)

Extensions

a(10)-a(35) from Robert Price, Jun 02 2012