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A213163 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(3,0,-,0)(x). 1

%I #30 Feb 19 2024 12:10:19

%S 1,7,32,122,422,1376,4315,13165,39360,115860,336876,969792,2768917,

%T 7851187,22130912,62066126,173294930,481976480,1335880495,3691245145,

%U 10171349376,27957706152,76672984152,209839988352,573211991977,1563112278751,4255708706720

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

%H Vincenzo Librandi, <a href="/A213163/b213163.txt">Table of n, a(n) for n = 4..1000</a>

%H 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 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-17,17,-7,1).

%F From _Vaclav Kotesovec_, Nov 25 2012: (Start)

%F a(n) = (7*n+6)*F(2*n)/5 - (4*n+5)*F(2*n+1)/5 + 1, where F is Fibonacci number (A000045).

%F Recurrence: a(n) = 7*a(n-1) - 17*a(n-2) + 17*a(n-3) - 7*a(n-4) + a(n-5).

%F G.f.: x^4/(1-7*x+17*x^2-17*x^3+7*x^4-x^5). (End)

%t CCC[t] = (1 - (1 - 4*t)^(1/2))/(2*t); NQ0[t, x] = ((1 + t - t*x) - ((1 + t - t*x)^2 - 4*t)^(1/2))/(2*t); NQ1[t, x] = 1/(1 - t*NQ0[t, x]); NQ2[t, x] = 1/(1 - t*NQ1[t, x]); NQ3[t, x] = 1/(1 - t*NQ2[t, x]); CoefficientList[Coefficient[Simplify[Series[NQ3[t, x], {t, 0, 20}]], x], t] (* Robert Price, Jun 06 2012 *)

%t CoefficientList[Series[1/(1 - 7*x + 17*x^2 - 17*x^3 + 7*x^4 - x^5), {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 25 2012 *)

%o (Magma) I:=[1,7,32,122,422]; [n le 5 select I[n] else 7*Self(n-1)-17*Self(n-2)+17*Self(n-3)-7*Self(n-4)+Self(n-5): n in [1..30]]; // _Vincenzo Librandi_, Nov 25 2012

%K nonn,easy

%O 4,2

%A _Robert Price_, Jun 06 2012

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)