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

%I A212342 #46 Feb 15 2025 15:49:20
%S A212342 1,1,2,5,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,
%T A212342 252,275,299,324,350,377,405,434,464,495,527,560,594,629,665,702,740,
%U A212342 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
%N A212342 Sequence of coefficients of x^0 in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).
%H A212342 S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I</a>, arXiv preprint arXiv:1201.6243 [math.CO], 2012. Sect 5.1 gives rational g.f.
%H A212342 Kyu-Hwan Lee, Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%F A212342 For n>=2, a(n)=(n^2+n-2)/2. - _Robert Price_, Jun 02 2012
%F A212342 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
%t A212342 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 *)
%Y A212342 Cf. A000096, A132337, A212346.
%Y A212342 A201163 is similar. - _Robert Price_, Jun 02 2012
%K A212342 nonn,easy
%O A212342 0,3
%A A212342 _N. J. A. Sloane_, May 09 2012
%E A212342 a(10)-a(35) from _Robert Price_, Jun 02 2012
%E A212342 Added a(0) to correspond to given offset and to be consistent with A212346, _Robert Price_, Jun 02 2012