A212338 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).

2, 7, 21, 53, 124, 273, 577, 1181, 2358, 4614, 8880, 16854, 31612, 58691, 108003, 197203, 357596, 644463, 1155059, 2059897, 3656988, 6465660, 11388480, 19990140, 34976870, 61019071, 106160481, 184228193, 318948124, 550962717, 949781269, 1634103701, 2806342578

Offset: 3

Author

N. J. A. Sloane, May 09 2012

Keywords

• nonn
• easy

Comments

Apparently the number of Dyck n-paths that have n-2 peaks after changing each valley to a peak by the transformation DU -> UD. E.g., the Dyck 3-paths UUUDDD and UUDUDD have 1 peak after changing DU to UD so a(3) = 2. - David Scambler, Sep 03 2012

Links

• Robert P. P. McKone, Table of n, a(n) for n = 3..5000
• S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012-2014.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Crossrefs

Cf. Column 2 of A091533. Partial sums of A036681.

Programs

Mathematica

LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 7, 21, 53}, {3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
A212338[1, n2_] = 0; A212338[n1_, 1] = 0; A212338[2, n2_] = 0; A212338[n1_, 2] = 0; A212338[3, 3] = 1; A212338[n1_, n2_] := A212338[n1, n2] = A212338[n1 - 1, n2] + A212338[n1, n2 - 1] + A212338[n1 - 1, n2 - 1] + A212338[n1 - 2, n2] + A212338[n1, n2 - 2]; Table[A212338[5, y], {y, 3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
QQQ2[t,x]=2/(1 + (t*x - t)*(1 + t) +((1 + (t*x - t)*(1 + t))^2 - 4*t*x)^(1/2)); CoefficientList[Coefficient[Series[QQQ2[t, x], {t, 0, 22}], x],t] (* Robert Price, Jun 05 2012 *)

Formula

g.f. -x^3*(2+x) / (x^2+x-1)^3, i.e., a(n) = 2*A001628(n-3) + A001628(n-4). - R. J. Mathar, Jun 27 2012

a(n) = a(n-1) + a(n-2) + A067331(n-3). E.g., a(5) = 21 = 7 + 2 + 12. - David Scambler, Sep 03 2012