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 A382574 #28 Apr 07 2025 09:46:58
%S A382574 1,1,2,5,13,35,90,225,525,1181,2526,5289,10729,21583,42566,83909,
%T A382574 163225,318713,616122,1198029,2309829,4483643,8635314,16750761,
%U A382574 32247973,62538517,120378518,233428337,449294497,871206887,1676835486,3251439501,6258092337,12134600945
%N A382574 a(n) is the number of shallow permutations of length n that avoid 123.
%C A382574 A shallow permutation is a permutation that satisfies the lower bound of the Diaconis-Graham inequality (i.e., so that the total displacement is equal to the sum of the length and reflection length).
%H A382574 Michael De Vlieger, <a href="/A382574/b382574.txt">Table of n, a(n) for n = 0..3494</a>
%H A382574 Kassie Archer, Aaron Geary, and Robert Laudone, <a href="https://arxiv.org/abs/2412.11999">Pattern-avoiding shallow permutations</a>, arXiV:2412.11999 [math.CO], 2024.
%H A382574 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-12,22,-12,-2,4,-1).
%F A382574 G.f.: (1-3*x+11*x^3-13*x^4+7*x^5+6*x^6+3*x^7) / ((1-x)^4 * (1-4*x^2+x^4)).
%t A382574 CoefficientList[Series[(1 - 3*x + 11*x^3 - 13*x^4 + 7*x^5 + 6*x^6 + 3*x^7)/((1 - x)^4*(1 - 4*x^2 + x^4)), {x, 0, 33}], x] (* _Michael De Vlieger_, Apr 01 2025 *)
%Y A382574 Cf. A062869, A301897.
%K A382574 nonn,easy
%O A382574 0,3
%A A382574 _Kassie Archer_, Mar 31 2025