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 A220902 #44 Sep 08 2022 08:46:04
%S A220902 2,4,11,33,104,339,1133,3861,13364,46852,166022,593674,2139552,
%T A220902 7763305,28337265,103981965,383351580,1419269280,5274495930,
%U A220902 19669409790,73580417040,276043317030,1038327097314,3915101867778,14795310818024,56028144245304,212581753906508,808027815817012
%N A220902 a(n) = Catalan(n) - A000245(n-2).
%H A220902 Robert Israel, <a href="/A220902/b220902.txt">Table of n, a(n) for n = 2..1667</a>
%H A220902 Filippo Disanto and Thomas Wiehe, <a href="http://arxiv.org/abs/1210.6908">Some instances of a sub-permutation problem on pattern avoiding permutations</a>, arXiv preprint arXiv:1210.6908 [math.CO], 2012 (see b_n).
%H A220902 F. Disanto and T. Wiehe, <a href="http://dx.doi.org/10.1016/j.disc.2014.06.028">On the sub-permutations of pattern avoiding permutations</a>, Discrete Math., 337 (2014), 127-141.
%F A220902 a(n) = Catalan(n) - Catalan(n-1) + Catalan(n-2). - _Andrei Asinowski_, Dec 16 2015
%F A220902 G.f.: (1 + x + x^2*C(x)^3)*x*C(x) - x where C(x) is the g.f. of A000108. - _Philippe Deléham_, Feb 04 2014
%F A220902 Conjecture: (n+1)*a(n) +(-5*n+3)*a(n-1) +(5*n-13)*a(n-2) +2*(-2*n+9)*a(n-3)=0. - _R. J. Mathar_, May 30 2014
%F A220902 From _Robert Israel_, Dec 31 2015: (Start)
%F A220902 Mathar's conjecture can be verified by expressing a in terms of factorials and simplifying.
%F A220902 G.f.: (1-3*x+x^2 -(1-x+x^2)*sqrt(1-4*x))/(2*x). (End)
%F A220902 E.g.f.: (1/6)*(-3*(3-x) + exp(2*x) * ( (9 -15*x +8*x^2)*BesselI(0, 2*x) - (6 -13*x +8*x^2)*BesselI(1, 2*x) ) ). - _G. C. Greubel_, May 03 2021
%p A220902 catalan:= n -> (2*n)!/n!/(n+1)!:
%p A220902 A220902:= n -> catalan(n) - catalan(n-1)+catalan(n-2):
%p A220902 map(A220902, [$2..100]); # _Robert Israel_, Dec 31 2015
%t A220902 Table[CatalanNumber[n] - CatalanNumber[n-1] + CatalanNumber[n-2], {n, 2, 30}] (* _Vincenzo Librandi_, Dec 24 2015 *)
%t A220902 CoefficientList[ Series[-x + (1 -Sqrt[1-4x])(1 -(-1 +Sqrt[1-4x])^3/(8x) +x)/2, {x, 0, 26}], x] (* _Robert G. Wilson v_, Dec 24 2015 *)
%o A220902 (Magma) [Catalan(n)-Catalan(n-1)+Catalan(n-2): n in [2..30]]; // _Vincenzo Librandi_, Dec 24 2015
%o A220902 (PARI) my(x='x+O('x^50)); Vec((1+x+x^2*((1-sqrt(1-4*x))/(2*x))^3)*x*((1-sqrt(1-4*x))/(2*x))-x) \\ _Altug Alkan_, Dec 24 2015
%o A220902 (Sage) [sum((-1)^j*catalan_number(n-j) for j in (0..2)) for n in (2..30)] # _G. C. Greubel_, May 03 2021
%Y A220902 Cf. A000108, A000245.
%K A220902 nonn,easy
%O A220902 2,1
%A A220902 _N. J. A. Sloane_, Jan 01 2013