A212346 -id:A212346 - cp's OEIS frontend

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

N. J. A. Sloane, May 09 2012

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.

Cf. A000096, A132337, A212346.

A201163 is similar. - Robert Price, Jun 02 2012

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 *)

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

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

A212347 Sequence of coefficients of x^1 in marked mesh pattern generating function Q_{n,132}^(0,4,0,0)(x).

Original entry on oeis.org

14, 56, 144, 300, 550, 924, 1456, 2184, 3150, 4400, 5984, 7956, 10374, 13300, 16800, 20944, 25806, 31464, 38000, 45500, 54054, 63756, 74704, 87000, 100750, 116064, 133056, 151844, 172550, 195300, 220224, 247456, 277134, 309400, 344400, 382284, 423206, 467324, 514800, 565800, 620494, 679056, 741664, 808500, 879750, 955604

Offset: 5

N. J. A. Sloane, May 09 2012

S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243, 2012
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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]); CoefficientList[Coefficient[Simplify[Series[QQ4[t, x], {t, 0, 35}]],x],t] (* Robert Price, Jun 04 2012 *)
Table[(n-4)*(n-3)*(n+1)*(n+2)/6, {n, 5, 50}] (* Jean-François Alcover, Sep 21 2017 *)

For n>=5, a(n)=(n^2-2n-8)(n^2-2n-3)/6 or a(n)=(n-4)*A212346(n-1).

G.f.: -2*x^5*(2*x^2-7*x+7) / (x-1)^5. - Colin Barker, Jul 22 2013

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

cp's OEIS Frontend

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

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