A212342 Sequence of coefficients of x^0 in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).

1, 1, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377, 1430, 1484, 1539, 1595, 1652, 1710, 1769, 1829, 1890, 1952, 2015, 2079

Offset: 0

Author

N. J. A. Sloane, May 09 2012

Keywords

• nonn
• easy

Links

• S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012. Sect 5.1 gives rational g.f.
• Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Crossrefs

Cf. A000096, A132337, A212346.

A201163 is similar. - Robert Price, Jun 02 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]); q=Simplify[Series[QQ3[t, x], {t, 0, 35}]]; CoefficientList[q /. x -> 0, t] (* Robert Price, Jun 04 2012 *)

Formula

For n>=2, a(n)=(n^2+n-2)/2. - Robert Price, Jun 02 2012

For n>=5, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). G.f.: (1-2*x+2*x^2+x^3-x^4)/(1-x)^3. - Colin Barker, Jul 06 2012

Extensions

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

Added a(0) to correspond to given offset and to be consistent with A212346, Robert Price, Jun 02 2012