This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A212338 #34 Jun 27 2022 08:24:59
%S A212338 2,7,21,53,124,273,577,1181,2358,4614,8880,16854,31612,58691,108003,
%T A212338 197203,357596,644463,1155059,2059897,3656988,6465660,11388480,
%U A212338 19990140,34976870,61019071,106160481,184228193,318948124,550962717,949781269,1634103701,2806342578
%N A212338 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).
%C A212338 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
%H A212338 Robert P. P. McKone, <a href="/A212338/b212338.txt">Table of n, a(n) for n = 3..5000</a>
%H A212338 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], 2012-2014.
%H A212338 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-5,0,3,1).
%F A212338 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
%F A212338 a(n) = a(n-1) + a(n-2) + A067331(n-3). E.g., a(5) = 21 = 7 + 2 + 12. - _David Scambler_, Sep 03 2012
%t A212338 LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 7, 21, 53}, {3, 35}] (* _Robert P. P. McKone_, Jan 14 2022 *)
%t A212338 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 *)
%t A212338 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 *)
%Y A212338 Cf. Column 2 of A091533. Partial sums of A036681.
%K A212338 nonn,easy
%O A212338 3,1
%A A212338 _N. J. A. Sloane_, May 09 2012